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What is the initial velocity and the acceleration of the train? A motorcyclist and a cyclist simultaneously begin to move from rest. The acceleration of the motorcyclist is three times greater than the acceleration of the cyclist By how many times will the velocity of the motorcyclist be greater? After the same interval of time as the cyclist. It then moves the next 12 km in 2 minutes. Calculate its acceleration, which is constant throughout the motion. When it approaches the next station, it stops by decelerating at 1.

If the train travels the distance between two stations in 2 minutes, what is the distance between the two stations in kilometres? On a straight road a lorry and a car are at rest at a fixed distance apart. After the lorry travels 96 m. How many seconds later will the car overtake the lorry?

How many metres is the car behind the lorry at the start? Take the initial position to be zero For both of the vehicles. When the distance between the vehicles is 75 ra K starts to slow down at a constant deceleration and, at the same instant, L starts to speed up at a constant acceleration towards K. If the speeds of the lorries become equal at point R how many metres is the distance MM? An object falls freely with an acceleration of 9.

What would the result be if the same experiment was carried out in an evacuated environment? If there were no air resistance, at what velocity would a rain drop strike the ground From a cloud of height m above the ground? A stone is released freely from a tower of height 80 m, How long does it take to reach the ground? At what velocity does ft strike the ground? Calculate its velocity and height after 2 s. An object, thrown upwards at v Qj reaches a maximum height of m. Calculate the initial velocity v Q. Calculate the total flight time.

Calculate Its total time of the flight Its velocity just before it strikes the sea. How many metres is the stone above sea level at t—3 s. How many metres high is the building? You can calculate a friend's reaction time in a simple experiment as follows: Hold a banknote or a ruler between your friend's fingers, just over his hand. Then ask your friend to catch it when you release it What is the maximum reaction time possible to catch a 16 cm banknote, as shown in the figure above?

Objects K and L are thrown at the same instant of time, as shown in the figure and meet at height hj. Two objects are released from the same height within a time interval of 1 s. An object is released from the balloon and falls freely If the object falls to the ground in 8 s T find The height oF the balloon from the ground at the moment the object is released. The velocity of the object at the moment it strikes the ground. What should the height, h, be so that both objects land at the same time?

Motion in One Dimension 5? However, the discussion was confined to one dimension only. In this chapter, the discussion will be extended to motion in a two-dimensional plane, where objects move in both x and y directions at the same time. Later on in this chapter relative motion of an object is described.

The parabolic path which is common to all objects in projectile motion is called a trajectory. See the examples from everyday life, as shown in Figure 3. Let s consider the projectile motion oF a ball t as shown in Figure 3. The origin of the x-y coordinate system is placed at the point from which the ball is thrown into the air.

Let's resolve the initial velocity of the ball v 0 Into its components along the x and y directions Figure 3. Figure 3,3 The position and velocity of a projectile at various instants of time. The time it takes to cover Its range is called the time of flight of the projectile. Motion in the y-direction: This is motion at a constant acceleration which is identical to that of a freely falling object under the effect of gravity. Examine the experiments shown in Figure 3Aa and Figure 3,4.

The equations derived in Chapter 2 are also valid here. Projectile motion is actually the superposition of two types of motion; One in the horizontal - Motion with constant velocity One in the vertical - Motion with constant acceleration The velocity of the projectile at any instant is the resultant vector of the rectangular components and v.

Since they have the same vertical acceleration, they strike the floor at the same time. Motion in Two Dimensions 59 The projectile motion of a golf boll Example 3. Calculate the time for the ball to reach its maximum height? What is the ball's velocity just before It strikes the ground? First select the direction of the vertical component of the initial velocity to be positive. In this selection, the acceleration becomes negative because it is always directed downwards. Find the following o The time of flight of the ball.

The distance away from the base of the building that the ball strikes the ground. The velocity of the ball just before it strikes the ground. The angle the ball makes with the horizontal after a time of 1 s. Solution Since the initial velocity of the ball is horizontal, this motion is the second half of the projectile motion trajectory of an object thrown upwards from ground level We can consider the initial height of the ball as the maximum height of a projectile thrown from the ground, where the initial vertical velocity was zero.

If the player Is 9 m away from the net when he throws the ball a How many seconds later does the ball pass through the basket? What is the height of the net from the ground? What horizontal distance from the location should the aeroplane release the package to land exactly where the mountaineers are trapped?

Calculate the velocity of the package just before It strikes the mountain. Mechanics - 3. On passing traffic lights, we ordinarily think of them as being stationary and ourselves in motion. Things can, however, we viewed from a different perspective if you imagine yourself as a stationary observer in the car, and observe the motion of the traffic lights.

Physically it is also possible to think of the car as being at rest and the traffic lights moving at the same speed but in the opposite direction! Next, think about a bus moving at the same velocity as the car, shown in. The motion of the bus with respect to the car appears to be stationary. So the motion of the car, the bus or the traffic lights depends on the measurement of the observer. In other words, all motion is relative to a given frame of reference. The velocity of A relative to B, in the form of subscripts, is written as and calculated by, AB Note that all velocity values in relative motion problems are considered to be constant.

Figure 3. Example 3. Find the velocities of the car and the motorcycle a A policeman, who is at rest next to the street b An observer in the car c A motorcyclist. Motion in Two Dimensions 63 utio All motions is in one dimension along a straight line. The velocity of the car is zero, li. The velocity of the motorcycle is zero. Similarly the velocity of A relative to a third frame of reference, C, is v AC and the velocity of the frame of reference C relative to that of B is These are related by the simple equation shown in the right-hand figure, Mote that the two outer subscripts on the right-hand side must be In the same order as the two subscripts on the left-hand side.

The two inner subscripts on the right-hand side must be the same, so they cancel. Mechanics Relative motion in one dimension Example 3. Solution a The train is stationary relative to the passengers. Relative Velocity in Two Dimensions The examples solved above are some applications of relative velocity only In one dimension. The objects In these examples are moving either in the same or in opposite directions relative to each other.

However, straight-line motion is rare in real life. Thus, another method must be used for analysing relative motion problems in real life, for example, for objects moving perpendicular to each other, as shown in Figure 3. Knowing that velocity is a vector, vector addition and subtraction can be used to solve such problems in two-dimensions. The resultant displacement of m the boat relative to the Earth in I s is 5. The range of a projectile is the horizontal distance covered by the projectile if it returns to its original height.

How do the acceleration and the horizontal and vertical velocities of the object change? When do the magnitudes of these quantities reach their maximum and minimum values? What is its time of in flight? It moves until it drops to the same horizontal level from which it was thrown. How would the time of flight, the range and the maximum height of an object exhibiting projectile motion change if a only the vertical component of its initial velocity is increased?

The girl in the picture is performing a long jump which is an example of projectile motion. Bob Beamon, an American athlete, set a long-standing world record for the long jump in with a jump of 8. His world record stood for 23 years, and was named one of the live greatest sports moments of the 20 th century. If Beamon left the ground at an angle of How many seconds later will the ball strike the ground? Three balls are sequentially thrown at the same instant from the top of a building, at the same speed. Ball 1 is thrown vertically upwards, bail 2 is thrown vertically downwards and ball 3 is thrown horizontally, as shown in the figure.

Do all balls reach the ground at the same Lime? If not, in what order do they fall? Do they strike the ground at the same speed? If not, list their velocities in order of magnitude. Do the masses of the balls affect the times they take to fall? How does the time of flight and the range of the object change, when the magnitude of the initial velocity is tripled? The ball passes over the defence players and the goal keeper catches if 2 m above the ground. What is the horizontal distance covered by the ball during its flight? Calculate the time it takes to complete its trajectory.

Calculate the velocity of the stone just before it strikes the ground. Calculate the angle the stone makes with the horizontal after 2 s. If it lands at a point cm away from the table a Find the time the plate takes to land on the floor. At what velocity does the plate strike the floor? A rifle is aimed horizontally at a point 60 m away, as shown in the figure. The bullet misses the target and strikes the wall 13 cm below the target point. If the effects of air resistance are neglected Calculate the time that the bullet remained in the air b Find the bullet's muzzle velocity. How many metres should the electron travel in the horizontal direction in order to have a declination of 4b cm? If the Fireman holds the pipe 1 m above the ground; How many seconds does it take for the water to reach the window?

What is the height of the fourth floor window from the ground? What is the velocity of the water at the moment it enters the window? Morion in Two Dimensions 69 the skier's final velocity at the moment of landing on the ground. Draw vertical and horizontal velocity-time graphs of the object. A child standing at the edge of a cliff at the seaside hits a ball into the sea t 16 m away from the cliff, with a stone, as shown into the figure. Acrobat B, who hangs by her knees on another trapeze, is stationary 3. Can acrobat B catch acrobat A?

H A ball is thrown at an initial velocity of v making an angle with the horizontal. At this minimum velocity what will the distance D, from the service position to the point at which the ball strikes the ground, be? A golf player strikes a golf ball so that ft has a range of 40 m T as shown in the figure. Find the ball's maximum height At what angle did he throw the ball? How many metres away is the target from the vertical projection of the point where the bomb is released, x?

What is the height of the bomb 5 s after being dropped? What type of motion does the bomb appear to have to the pilot? At what horizontal distance from point L on the car does the object strike the car? If the gymnast catches the ball when he arrives at point L What type of motion does the ball exhibit relative to the Earth?

Find the time the object is in the air. What is the velocity of the gymnast? A bail is thrown horizontally and photographed using a multi-flash camera. Its motion is shown sequentially in the right-hand figure. The time interval between each snapshot is 0. The graph shows how its vertical component of velocity changes in time.

What is the horizontal distance travelled by the stone? At the moment a red ball is thrown from point K with a velocity of v, a green ball is released from point L, as shown in the figure. The balls collide at point M What is the initial velocity v of the red bail? Mote that the red ball always strikes the green ball regardless of its initial velocity provided that both are thrown and released at same time. Can you explain why?

What is the velocity of the van relative to the bus? That is the velocity relative to the river. What is the velocity of the fish relative to the Earth and the river a ] If it swims in the same direction as the river? A bus travels at a constant velocity. If a child in the bus jumps upwards from rest T where does she land relative to the point from which she jumped on the bus? If the bus was accelerating, what would your answer be?

What is the velocity of the aeroplane relative to the Earth?

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Two canoe riders who are able to ride at equal velocities on still water are sailing their canoes on a constantly flowing river. One is sailing in the same direction as the water flow and the other is sailing in the opposite direction to the flow. At what velocity does the motorcyclist observe the car moving at?

The pilot of a helicopter, which flies due north at a velocity of 3v relative to the Earth, observes an aeroplane flying at a velocity of 4v due east What is the velocity of the plane relative to the ground? What is the magnitude of the velocity of the rain drops a Relative to the train? How far from point B does the boat drift?

What will the distance between the boats be when they reach the other side of the river? A passenger in the first train observes that the second train passes his train in a time interval of 6 seconds. What is the length of the second train?

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In a given escalator it takes 30 s to move up to the next floor. When the escalator is out of service, a person walking up the stairs at a constant velocity takes 45 s to climb up to the next floor. How long will it take if he walked at the same constant velocity on a moving escalator? Motion in Two Dimensions 75 The Laws of Motion In Chapter 2 The motion of objects was described with the quantities of kinematics such as displacement, velocity, and acceleration.

However, these quantities cannot describe ail aspects of motion, for example what starts objects moving and what causes change in their motion. In this chapter the causes and factors affecting motion will be discussed, using the concept of force and mass. The three fundamental iaws of motion, discovered and :om: dated by Isaac Newton will be discussed. These laws explain rest, constant motion, accelerated motion, and they also describe how baianced and unbalanced forces behave , and thus give rise to various states of motion.

The branch of physics dealing with the causes of motion is called dynamics. When a car is pushed, as shown in Figure 4,1. When a locomotive pulls a train, as shown in Figure 4. Lb, it exerts a force on the train. When a ball is kicked, as shown in Figure 4. C, a force Is exerted upon it. When a hammer strikes a nail,, or a motor pulls a lift, or the wind blows the leaves of a tree, a force is being exerted. Such a force between two objects which are In contact is called a contact force. There are also forces, called field forces, which act even when objects are separated by a distance.

Gravitational force, magnetic force and electric force are known as field forces. When an object is released near the surface of the Earth, it falls down towards the surface of the Earth due to the gravitational force also called the weight of the object between the object and the Earth, which are not originally in contact, as shown in Figure 4. Two magnets repel each other when their same poles are brought closer or attract each other when their opposite poles are brought closer, even though the magnets are not in contact, as shown in Figure 4.

The attractive electric force between two like charges and the repulsive electric force between two unlike charges, as shown in Figure 4. Lf is an example of a field force. Figure 4. The magnitude or strength of a force can be measured by a spring mechanism called a dynamometer or a spring balance. In a dynamometer, a spring is fixed to one end of the mechanism. When a force is applied to the other end of the spring, the spring extends in proportion to the applied force. The greater the force, the greater the extension.

The dynamometer scale is calibrated using known forces the weights of standard masses , as shown in Figure 4. Since force is a vector quantity, it has a direction as well as a magnitude and Is represented by an arrow whose length is proportional to the magnitude of the force. The SI unit of force is the Newton and is abbreviated by N. Figure 4,2 The forces applied upon the gymnast by the ropes are contact forces and the gravitational force exerted upon him by the Earth is a field force. The Laws of Motion 77 4. Forces are added by the rules of vector addition, as shown in Figure 4.

Gravitational force acting on a child force applied by the engine of an aeroplane Engine force of a car which has started to me Force between the nucleus and the electron of a hydrogen atom Q nucleus w electron Force applied by engine of a train Gravitational force between the Earth and the Moon Moon 3 ; 8 3. From this simple experiment the first law of motion can be inferred, it was stated by Mewton more than 3 centuries ago as follows: If the net force acting on an object is zero ti If it is at rest, it will stay at rest.

Inertia Inertia is the tendency of an object to resist any change in its state of motion. If an object is at rest, it tends to remain at rest. The following situations can be discussed in order to understand the law of inertia. If you are in a car which is at rest. Since your body resists the change in its resting state, you experience and feel this force.

This Force overcomes your inertia and puts you in motion with the same velocity as the car. Your body resists the change in the direction of its velocity due to your inertia and tries to keep moving on the straight line. And you Feel as though you are being pushed to the left. Here are some other examples of inertia; : When a hard surface is struck with the back of a hammer, it stops suddenly and the hammer head feels tightened.

When a sheet of kitchen paper is pulled slowly more and more paper rolls off. However, when a sheet is pulled quickly, it is tom off the roll since the pulling force doesn't have enough time to overcome inertia. Unless a net force acts on it, it remains at rest.

If the object initially were moving with a constant velocity, it would continue to move with the same constant velocity until it experiences a net force. Mass is a Measure of Inertia Mass is a measure of the response of an object to an external force. The greater the mass of an object, the greater the inertia and the less that object accelerates changes its state of motion under the action of a net force. From this example a relationship can be set-up between the accelerations and the masses of the objects experiencing the same net force.

If one of the masses in this equation is known, the unknown mass of an object can be found after the accelerations are measured. Finally, mass is a scalar quantity, it has no direction, it is always positive. In that case, the object either remains at rest or moves in a straight line with a constant speed. The second law of motion explains what happens when the net force acting on an object is not zero. As stated in the simple experiment of the previous section, a net force acting on a an object causes it to accelerate.

In this case, the net force acting on the object is F and the object gains an acceleration a, IF Figure 4. If the applied force of mass m to accelerate at a value of a, is tripled, the acceleration triples and so on. From such experiments it can be concluded that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

From all these experiments, the second taw of motion is stated as follows The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This law was discovered and formulated by Newton as 80 Mechanics Note that net force and acceleration always have the same direction. Weight We are well aware that objects are attracted to the Earth. The force exerted by the Earth on an object is called the weight of that object. This force is directed towards the centre of the Earth and its magnitude changes with location.

At sea level the Earth exerts a force of 9. Experiments show that in the same location the ratio of the weight of any object to its mass gives the same value, called the gravitational field strength, g. We can obtain the mass of an object in two different ways 1. The object can be given an acceleration, a, by exerting a net force, F nct on it and these quantities measured. Then the ratio of the net force to the acceleration, from the second law of motion, gives the mass of the object, which is called inertial mass.

The weight, w of the object can be measured with an equal arm balance. They were observed to be the same. The Laws of Motion Here it is observed that the acceleration of a freely falling body and gravitational field strength are equal Due to this, gravitational field strength is also called gravitational acceleration. Inertial Reference Frames An inertial reference frame is a coordinate system in which the first law of motion is valid: in this coordinate system, an object at rest remains at rest and an object in motion with a constant velocity continues to move with the same velocity, unless an unbalanced force net force acts on it.

An inertial reference frame is a coordinate system which does not accelerate. An inertial reference frame can be explained by considering the following situation. Imagine that you are standing still on roller skates on a bus. Obviously your acceleration and the net force acting upon you is zero, relative to the bus. Thus, the first law of motion is valid in the bus. Thus, the bus at rest can be accepted to be an inertial reference frame. Relative to the bus, your acceleration and the net force acting upon you is zero. Therefore, the first law of motion is valid on the bus.

The bus moving with constant velocity can be accepted to be an inertial reference frame. You and the bus are initially at rest, then the bus accelerates forwards, as shown in Figure 4. Relative to the bus frame you are accelerating backwards. It Is the bus which is acceierating.

Relative to the bus, you were initially at rest, then you accelerated backwards, although you did not experience any net force. So the accelerating bus cannot be accepted to be an inertial reference frame because the first law of motion is not valid in it. As a result, any reference frame the bus in the examples above which is at rest or in motion with a constant velocity relative to another inertial reference frame the Earth in the examples above is itself an inertial reference frame.

The laws of motion are valid only in inertial reference frames. Actually the Earth cannot be an inertial reference frame. It has centripetal acceleration due to its rotational motion about its own axis and an orbital motion around the Sun. In general, it is very difficult to define an inertial reference frame in the universe, since everything is in a state of motion.

However, since the distant stars can be assumed to be at rest, the systems which move with a constant velocity relative to these stars can be accepted to be inertial reference frames. Find a the acceleration of the load, b the speed which the load reaches 5 s later. From the equation of the second law of motion we can find its acceleration. Forces always occur in pairs. Thus, Newton stated the third law of motion as follows When one object exerts a force on a second object, the second object exerts an equal but opposite force on the first.

The force that the first object exerts on the second is called an action force and the force that the second object exerts on the first is called a reaction force. This law is sometimes expressed as To every action there is an equal and opposite reaction. These forces have the following properties Action-reaction pairs are equal in magnitude, but opposite in direction, and they act along the same line.

For example, the force exerted by the Earth on an apple is the weight of the apple, w. The reaction to this force is the force exerted by the apple on the Earth, - w, where w and - w are equal in magnitude see Figure 4. Due to the force w, the apple accelerates towards the Earth, Also the Earth accelerates toward the apple due to the reaction force, - w. However, since the Earth has a huge mass, its acceleration Is negligibly small. If two forces which are equal in magnitude and opposite in direction act on the same object as in Figure 4.

Therefore these forces do not cancel each other out, since they act on different objects. Figure 4,16 The tyres of a car push the ground backwards; the ground then pushes the tyres forwards. However, although action and reaction forces are equal in magnitude and opposite in direction, they do not cancel each other out, because they act on different objects.

For example, when a force, F is exerted on a grocery cart by a shopper, as in Figure 4,13 the cart reacts on the shopper with a force of -F Since F acts on the cart and -F acts on the shopper, the cart moves. Action-reach on pairs are of the same type; either two contact forces or two Reid forces.

Action-reaction pairs act for the same time interval. Here are some simple experiments of the third law of motion Assume that you and a friend are standing on ice skates facing each other, as shown in Figure 4. If you push your friend away from you, you will observe that as he moves away from you, you will also move backwards because of his equal and opposite reaction force upon you.

You can only walk due to the reaction force of the ground on your shoes when you yourself exert an action force on the ground, as shown in Figure 4. The tyres of a car push on the ground, thus, the ground pushes on the tyres, as shown in Figure 4. Rockets also use the action and reaction principle. Thus, the rocket moves forwards. Prior to analysing the applications, the basic types of forces which will be useful in analysis will be examined.

Normal Contact Force Obviously, the acceleration of an object which is at rest on the Earth is zero, as shown in Figure 4. Here, we should emphasize that although the force of gravity, w and the normal contact force, N are equal in magnitude and opposite in direction, they are not action and reaction forces because they act on the same object, thus, they balance each other. As shown in Figure 4. In order to analyse these forces dearly, the object and the Earth are separated, as shown in Figure 4.

Tension force acts along the string and always pulls. The ropes and the strings in the problems are assumed to be massless. Therefore, the tension in a massless rope has the same value at all points along the rope. This fact will be proven in example 4. Thc Laws of Motion 85 Figure 4. Practically tension force can be measured as follows: Two adjacent particles a and b of a string, used to suspend an object from the ceiling are shown in Figure 4.

If the string is cut by separating the particles a and b and then a dynamometer placed between them, as shown in Figure 4. C, the value read by the dynamometer is the same as the magnitude of the tension force, T between the particles. Internal and External Forces In order to apply the laws of motion to an object or a system, the Forces need to be classified as external or internal forces. The forces acting on an object or a system from the environment are called external forces. As in Figure 4. Forces acting between the components of a system are called internal forces.

The tension forces between the objects of a system, as shown in Figure 4. When the laws of motion are applied to an object or a system, only the external Forces acting on them from the environment are of interest Since internal forces for example, tension forces as in Figure 4.

However, if the second law of motion is applied to each object of the system separately as shown in Figure 4. Some free-body diagrams of applied forces are shown below. Also, do not show the forces exerted by an object or a system on its environment in drawing the free body diagram. Select a suitable inertial reference frame.

Position the origin and the orientation of the coordinate axes and find the components of the forces along these axes. If there are more than one unknown, there must be the same number of equations as the number of unknowns. The unknowns, can be obtained by solving the equations. The net force acting on the system is force F. Since the weights of the objects and the normal contact forces on the objects are perpendicular to the direction of motion, they do not affect the motion.

Also,since the tension in the rope is an internal force, it is not taken into account. Since the system accelerates in the direction of F, the net force acting on mass m 2 is F-T The vertical forces do not affect mass m 2. Applying the second law of motion to the system, its acceleration is obtained. Calculate the acceleration he will gain b the time taken for him to reach point L. We assume that the surface is frictionless. When a body slides over the surface of another, its motion is always opposed by a retarding force that resists this motion.

This force is called the force of friction. Forces of friction are very important to us in our everyday lives. Without friction we could not walk, move, stop or turn comers. We would not be able to hold objects, for example a pencil, even if we could hold it, it would not write, since writing also relies on friction, we would not be able to drive cars. Suppose there is a block at rest on a horizontal surface, as shown in Figure 4. Before applying any force on the block, no force of friction acts upon It When the block is pulled with a small horizontal force F it still remains at rest.

From the second law of motion, it can be concluded that there must be another force acting on the block which opposes and balances the applied force F This force is the friction force, f exerted on the block by the surface. If the applied force is increased, the friction force acting on the block also increases. By decreasing this force it Is possible to keep the block in motion with a constant velocity a-Q , as shown in Figure 4.

Thus, the force required to start the motion of the block is slightly greater than the force required to keep it in uniform motion. We can summarise the experimental observations with Figure 4. Experiments have revealed the following empirical rules of friction.

Friction forces are always parallel to the surface of contact and opposite to the direction of motion or intended motion. Some average values of and p k for different pairs of materials are shown in Table 4.

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Generally ji k is less than p s. That is why it is easier to keep the block moving than to start it moving. The coefficient of kinetic friction is quite independent of the relative speed of the surfaces in contact The coefficients of friction are nearly independent of the contact area between the surfaces. Wficit Causes Friction? It actually occurs due to the bonds formed between the molecules at high points on the surfaces ho matter how smooth the surfaces of objects appear to be r in reality they are rough and they have irregularities on their surfaces, as shown in Figure 4.

However, the friction does not arise from only mutual contact of irregularities on the surfaces. Experiments show that the real causes of friction are The electrostatic forces" which occur between the molecules of the surfaces. When the flat surfaces of two objects are placed in contact, the actual microscopic area of contact is much smaller than the apparent macroscopic area of contact. It is rather like turning Switzerland upside down and placing it on top of Austria. Only the tips of mountains will touch. The actual area of contact is proportional to the normal force.

When the normal force is constant, actual area of contact is also constant. The actual contact area remains the same even when the apparent contact area is reduced because increased normal force per unit actual area brings more molecules closer to interact. That is to say, the number of molecules forming bonds between the surfaces in both cases of having small and large apparent contact area is the same. Consequently friction force is the same. As a block slides over a surface, the bonds between the surfaces form and break.

Up to a certain extent, smoothing a surface removes the irregularities and decreases friction. However, it brings more molecules, capable of bonding, closer and in this way It actually increases friction. The coefficient of kinetic friction between the block and the surface is XL If a force of 10 H is applied to the block, as shown in the figure. Find the friction force acting on the block. The block does not accelerate along the y axis. Applying the second law of motion to the y axis, the normal contact force can be found, and then the frictiona] force.

Prove that the tension in a string has the same value at all points along the string. Eventually the air drag balances her weight and she attains a terminal velocity. Terminal Velocity So far we have studied the motions of objects in the absence of air resistance. Air resistive forces acting on freely failing objects have not been considered. Actually any object in motion experiences an air resistive force commonly called air drag arising from collisions between the air particles and the moving object.

This results in an increase in the air drag to a value greater than her weight thus, her velocity decreases. The skydiver continues to descend at this constant velocity, as shown in Figure 4. Example 4. The gravitational force acting upon the person is called actual weight.

The normal contact force is called apparent weight. That is, the force an object exerts on a surface is its apparent weight. Similarly an object suspended with a string is acted upon by two forces: The downward force of gravity and the tension from the string. The force of gravity is the actual weight, whereas, the tension force is the apparent weight of the object Figure 4.

Let's analyse the apparent weight of a bag in a failing elevator. This reading T can be calculated by applying the second law of motion. As the elevator rises with an acceleration of g, as in Figure 4, The bag seems weightless. This phenomenon is called weightlessness. Summary The first law of motion states that If the net force acting on an object is zero If it is at rest, it will remain at rest.

Normal Vector of a Plane

The second law of motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The equality holds only when f has its maximum value. The force of kinetic friction is given by where f k is the magnitude of the force of kinetic friction, is the The force exerted by the Earth on an object is called the weight of that object it is directed towards the centre of the Earth and its magnitude changes with location.

It is equal to the product of the mass of the object and the gravitational acceleration g at that location. Thus, an isolated Force can never exist in nature. If no net force acts on a body, is it possible for it to 4. Explain the second law of motion 2. What is a "net force' 1? The magnitudes and the directions of some forces acting on an object on a frictionless surface are shown in the figure.

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5. Find the net force acting on the object. Explain the first law of motion and inertia 6, If a body is stationary, can we say that there is no force acting on it? Express the unit Newton in terms of base SI units. If there is only one force acting on a body, can it be at rest? Is the motion of bodies always in the same direction as that of the net force? Can the direction of the net force acting on an object be opposite to that of its acceleration?

To which law of motion is the operation of car seat belts related? Which forces act on an apple when it remains on a tree? According to the third law of motion, when you push somebody, he does not have a right to complain about you. What is the net force on the object?

What is the average force on the bullet while it is in the rifle? A man weighs N on Earth. Find the acceleration of the object shown in the figure. If the 3-kg box is pulled by a force of 18 M, as shown in the figure, find the acceleration of the boxes the tension in the rope between the boxes. Two perpendicular forces act on a g object. An object of mass lg is moving under a force of IN. Find its acceleration. The system in the figure moves under the effect of a force of 12 M on a smooth surface.

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Two blocks, of mass 5 kg each, are attached to two ends of a dynamometer by the strings shown in the figure. What is the force displayed on the dynamometer? If the system is released from rest, find the acceleration of the system, the tension in the string. They are placed on an inclined frictionless plane, as shown in the figure, then released. Find the acceleration gained by the system b the tension in the string. If the system shown in the figure is frictionless, what is the acceleration of the system?

For the frictionless system shown in the figure, find the acceleration of the system the tension T in the string. If the masses are at a height of 1. What is the velocity of the smaller mass when the greater one strikes the ground? If this experiment were carried out on the Moon, how many seconds would it take for the 2. If the pulleys and the surface are smooth what are the accelerations of m ] and m 2? If the coefficient of kinetic friction between the object and the surface is 0. If the coefficient of kinetic friction between the surface and the object is 0,15, find the acceleration of the object?

What is the coefficient of kinetic friction? An ice skater moving at 10 nVs slides to a halt in m on an icy surface. What is the coefficient of kinetic friction between the ice and the skates? For masses and m 2 to be able to stay stationary on m 3 what should the acceleration of m 3 be? A smooth Frictionless system is shown in the figure. In the system shown in the figure, the coefficient of friction between mass m 1 and the horizontal surface is 0. What should the mass of object M be so that nri j does not slide over the table? What would the acceleration of the system be if object M is removed?

Explain actual weight and apparent weight. A kg boy is standing on a scale in a lift, as shown in the figure. When the pendulum suspended from the roof of the vehicle comes to equilibrium, what will the tension in the string be? Find the accelerations of these masses according to an observer looking at the lift from outside. A parachutist has a mass of 60 kg and the mass of his parachute is 20 kg. The plain how a small force can sometimes effect These two conditions for equilibrium M. An object moving or rotating at a constant rate is in dynamic equilibrium. For an object to be in equilibrium, two conditions must be Fulfilled.

First Coiidlfton for Equilibrium For an object to be in equilibrium, the resultant force vector sum of all forces on it must be zero. Is a rest, as shown in the figure. Find the tension in the cord. Solution If the object stays at rest, the forces on it are in equilibrium. Find the tension in the string and the weight of the object. These forces are; the weight of the object w , the applied force F and the tension T in the string.

Using the first condition for equilibrium on the y axis, weight w can be obtained; The forces acting on the object can be transferred to the xy coordinate system and then the first condition for equilibrium can be applied to the object on both the x and y axis. However, with a spanner the screw can be tightened easily, as shown in Figure 5. The spanner may produce a large turning effect. The turning effect of a force is called the torque and is denoted by the Greek letter x. Torque is also called the moment of a force. The point about which a force tends to rotate an object is called the pivot turning point.

In Figure 5. The same Force can produce different torques on an object, depending on the point of application and the orientation. Figure 5. In all four cases in Figure 5. The torque acting in Figure 5. This is because the contribution of the parallel component of force to the torque is zero. Example 5. Find the torque produced by this force. Let us find its component along the y axis, F y which is perpendicular to the spanner. F y causes clockwise rotation. Consider the door in Figure 5.

Two forces, F 1 and F 2 , act on the door producing turning effects in opposite senses of rotation. Taking the anti-clockwise sense of rotation as positive and the clockwise sense of rotation as negative. The net torque is the sum of the individual torques Torque and Equilibrium If the net torque on the object is negative, then the body starts to rotate in the negative direction. This means that, to prevent the door from rotating, a clockwise torque of 31 Mm must act on it The minimum force to produce this torque must be applied perpendicularly, to the furthest point from the fulcrum.

Consider the object in Figure 5. Second condition for equilibrium For an object to be in equilibrium, the resultant torque sum of all torques acting on it must be zero. The object does not accelerate, but starts to rotate. It does not start to rotate, but it accelerates. A weightless beam is hinged at point A and suspended by a rope at point B to a wall. A N object is suspended from the midpoint If the beam is in equilibrium, as shown in the figure find a the tension T in the rope, b the reaction force R acting upon the beam exerted by the wall at point A.

The torques produced by R x and are zero because they are exerted on the beam at the point of rotation. The net torque is produced by the forces T and w. The box is subject to two forces F and as shown in the figure. If F 1 is 10 N and each side of the square is 1 m, find the force P 2 required for the box to be in equilibrium. Thus a pivot point at any place suitable for the problem can be chosen.

While the painter is at the position shown in the figure, what are the tensions T 1 and T 2 in the ropes? If the three forces are to be replaced by a single force such that the net force and the net torque on the object, with respect to the given pivot, remains constant what is the magnitude and point of application of this single resultant force?

The Centre of Gravity Since gravitational force always acts towards the centre of the Earth, gravitational forces affect all the particles exerting forces upon the particles in the same direction parallel to each other, as shown in Figure 5. The resultant of these forces produces the weight of the body. The application point of this resultant force is called the centre of gravity of the body.

A barbell with different weights on each end is placed in the xy coordinate system In Figure 5, The barbel! The centre of mass is a point where the whole mass of a body is assumed to be concentrated. In an environment where the gravitational field is uniform, the centre of gravity and mass are at the same point In gravity-free environments, as there is no weight, there is only the centre of mass. If the masses of the objects are 3 kg, 2 kg and 1 kg, respectively, find the coordinates of the centre of mass of the system formed by these objects.

The intersection points of the extension fines through these points is the centre of gravity of the object. The centre of gravity of bodies can be found using this fact. The cone shown in Figure 5. The forces on the cone are balanced. The centre of gravity is over the base. Its wide base and lower centre of gravity makes the cone more stable. A small push will cause it to lose its equilibrium position and small force applied to the cone will cause a torque, and the cone will fall. Wherever it lies, its centre of gravity will be towards the contact area. An object moving or rotating at a constant rate is In dynamic equilibrium.

For an object to be in equilibrium, two conditions must be fulfilled. For an object to be In equilibrium, the resultant force vector sum of all forces on it must be zero. The turning effect of a force is called torque and is denoted by the Greek letter t. An object may not be in equilibrium even though the first condition is fulfilled. The net force on an object may be zero but the object may not remain at rest. Therefore for an object to be in equilibrium, one more condition is needed.

For an object to be in equilibrium, the resultant torque sum of all torques acting on it must be zero. The tension in the rope is 30 M when an object of mass m is in equilibrium, as shown in the figure. What is mass, m r in kilograms? An object of mass m is in equilibrium with the aid of a horizontal force of 30 N.

What is the mass of the object, in kilograms? What is the tension force in the string, in Newtons, if the reaction force applied to the object by the inclined plane is 30 N? Two identical spheres having equal weights of 30 N are placed in a rectangular container, as shown in the figure. R 3 which are exerted on the spheres by the walls. Neglect friction, 6. The system in the figure is at rest. A force of 50 N is applied to a 0.

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