• Introduction

• Aristotle’s Fallacy

• The Law of Inertia

• Inertia

• Newton's First Law of Motion

• Newton's Second Law of Motion

• Momentum

• Impulse

• Aristotle’s Fallacy

• The Law of Inertia

• Inertia

• Newton's First Law of Motion

• Newton's Second Law of Motion

• Momentum

• Impulse

• We saw that uniform motion needs the concept of velocity alone whereas non-uniform motion requires the concept of acceleration in addition. So far, we have not asked the question as to what governs the motion of bodies. In this chapter, we turn to this basic question.

• Let us first guess the answer based on our common experience. To move a football at rest, someone must kick it. Clearly, some external agency is needed to provide force to move a body from rest.

• Likewise, an external force is needed also to retard or stop motion. You can stop a ball rolling down an inclined plane by applying a force against the direction of its motion.

• In these examples, the external agency of force is in contact with the object. This is not always necessary.

`"Example-"` A stone released from the top of a building accelerates downward due to the gravitational pull of the earth. A bar magnet can attract an iron nail from a distance.

• In short, a force is required to put a stationary body in motion or stop a moving body, and some external agency is needed to provide this force. The external agency may or may not be in contact with the body.

• Let us first guess the answer based on our common experience. To move a football at rest, someone must kick it. Clearly, some external agency is needed to provide force to move a body from rest.

• Likewise, an external force is needed also to retard or stop motion. You can stop a ball rolling down an inclined plane by applying a force against the direction of its motion.

• In these examples, the external agency of force is in contact with the object. This is not always necessary.

`"Example-"` A stone released from the top of a building accelerates downward due to the gravitational pull of the earth. A bar magnet can attract an iron nail from a distance.

• In short, a force is required to put a stationary body in motion or stop a moving body, and some external agency is needed to provide this force. The external agency may or may not be in contact with the body.

• The Greek thinker, Aristotle (384 B.C– 322 B.C.), held the view that if a body is moving, something external is required to keep it moving. According to this view, for example, an arrow shot from a bow keeps flying since the air behind the arrow keeps pushing it.

• Aristotelian law of motion may be phrased thus: An external force is required to keep a body in motion.

• Aristotelian law of motion is flawed, as we shall see. When the car is in uniform motion, there is no net external force acting on it but the opposing forces such as friction (solids) and viscous forces (for fluids) are always present in the natural world. This explains why forces by external agencies are necessary to overcome the frictional forces to keep bodies in uniform motion.

• To get at the true law of nature for forces and motion, one has to imagine a world in which uniform motion is possible with no frictional forces opposing. This is what Galileo did.

• Aristotelian law of motion may be phrased thus: An external force is required to keep a body in motion.

• Aristotelian law of motion is flawed, as we shall see. When the car is in uniform motion, there is no net external force acting on it but the opposing forces such as friction (solids) and viscous forces (for fluids) are always present in the natural world. This explains why forces by external agencies are necessary to overcome the frictional forces to keep bodies in uniform motion.

• To get at the true law of nature for forces and motion, one has to imagine a world in which uniform motion is possible with no frictional forces opposing. This is what Galileo did.

Galileo performed some experiments to understand the motion of objects.

`"Experiment 1"`

Galileo studied motion of objects on an inclined plane. During experiment he found that

(i) Objects moving down an inclined plane accelerate.

(ii) Objects moving up retard.

(iii) Motion on a horizontal plane is an intermediate situation.

Galileo concluded that an object moving on a frictionless horizontal plane must neither have acceleration nor retardation, i.e. it should move with constant velocity (Fig. 5.1(a)).

`"Experiment 2"`

• Another experiment by Galileo leading to the same conclusion involves a double inclined plane.

• A ball released from rest on one of the planes rolls down and climbs up the other. If the planes are smooth, the final height of the ball is nearly the same as the initial height (a little less but never greater). In the ideal situation, when friction is absent, the final height of the ball is the same as its initial height.

• If the slope of the second plane is decreased and the experiment repeated, the ball will still reach the same height, but in doing so, it will travel a longer distance. In the limiting case, when the slope of the second plane is zero (i.e. is a horizontal) the ball travels an infinite distance. In other words, its motion never ceases. This is, of course, an idealised situation (Fig. 5.1(b)).

• In practice, the ball does come to a stop after moving a finite distance on the horizontal plane, because of the opposing force of friction which can never be totally eliminated. However, if there were no friction, the ball would continue to move with a constant velocity on the horizontal plane.

• Galileo thus, arrived at a new insight on motion that the state of rest and the state of uniform linear motion (motion with constant velocity) are equivalent. In both cases, there is no net force acting on the body.

• It is incorrect to assume that a net force is needed to keep a body in uniform motion. To maintain a body in uniform motion, we need to apply an external force to encounter the frictional force, so that the two forces sum up to zero net external force.

`"Inertia"`

• If the net external force is zero, a body at rest continues to remain at rest and a body in motion continues to move with a uniform velocity. This property of the body is called inertia.

• Inertia means ‘resistance to change’.

• A body does not change its state of rest or uniform motion, unless an external force compels it to change that state.

`"Experiment 1"`

Galileo studied motion of objects on an inclined plane. During experiment he found that

(i) Objects moving down an inclined plane accelerate.

(ii) Objects moving up retard.

(iii) Motion on a horizontal plane is an intermediate situation.

Galileo concluded that an object moving on a frictionless horizontal plane must neither have acceleration nor retardation, i.e. it should move with constant velocity (Fig. 5.1(a)).

`"Experiment 2"`

• Another experiment by Galileo leading to the same conclusion involves a double inclined plane.

• A ball released from rest on one of the planes rolls down and climbs up the other. If the planes are smooth, the final height of the ball is nearly the same as the initial height (a little less but never greater). In the ideal situation, when friction is absent, the final height of the ball is the same as its initial height.

• If the slope of the second plane is decreased and the experiment repeated, the ball will still reach the same height, but in doing so, it will travel a longer distance. In the limiting case, when the slope of the second plane is zero (i.e. is a horizontal) the ball travels an infinite distance. In other words, its motion never ceases. This is, of course, an idealised situation (Fig. 5.1(b)).

• In practice, the ball does come to a stop after moving a finite distance on the horizontal plane, because of the opposing force of friction which can never be totally eliminated. However, if there were no friction, the ball would continue to move with a constant velocity on the horizontal plane.

• Galileo thus, arrived at a new insight on motion that the state of rest and the state of uniform linear motion (motion with constant velocity) are equivalent. In both cases, there is no net force acting on the body.

• It is incorrect to assume that a net force is needed to keep a body in uniform motion. To maintain a body in uniform motion, we need to apply an external force to encounter the frictional force, so that the two forces sum up to zero net external force.

`"Inertia"`

• If the net external force is zero, a body at rest continues to remain at rest and a body in motion continues to move with a uniform velocity. This property of the body is called inertia.

• Inertia means ‘resistance to change’.

• A body does not change its state of rest or uniform motion, unless an external force compels it to change that state.

• Newton built on Galileo’s ideas and laid the foundation of mechanics in terms of three laws of motion that go by his name. Galileo’s law of inertia was his starting point which he formulated as the First Law of motion.

`"First Law of Motion"`

Every body continues to be in its state of rest or of uniform motion in a straight line unless compelled by some external force to act otherwise.

• The state of rest or uniform linear motion both imply zero acceleration.

• If the net external force on a body is zero, its acceleration is zero. Acceleration can be non zero only if there is a net external force on the body.

`"Example-"` A spaceship out in interstellar space, far from all other objects and with all its rockets turned off, has no net external force acting on it. Its acceleration, according to the First Law, must be zero. If it is in motion, it must continue to move with a uniform velocity.

`"First Law of Motion"`

Every body continues to be in its state of rest or of uniform motion in a straight line unless compelled by some external force to act otherwise.

• The state of rest or uniform linear motion both imply zero acceleration.

• If the net external force on a body is zero, its acceleration is zero. Acceleration can be non zero only if there is a net external force on the body.

`"Example-"` A spaceship out in interstellar space, far from all other objects and with all its rockets turned off, has no net external force acting on it. Its acceleration, according to the First Law, must be zero. If it is in motion, it must continue to move with a uniform velocity.

Q 1360178915

An astronaut accidentally gets separated out of his small spaceship accelerating in inter stellar space at a constant rate of `100 m s^(−2)`. What is the acceleration of the astronaut the instant after he is outside the spaceship assuming there are no nearby stars to exert gravitational force on him?

Newton's first law states that when viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.

By the first law of motion, the acceleration must be zero.

Given there are no near by starts to exert gravitational pull, and the small spaceship exerts negligible gravitational attraction on him, the net force acting on the astronaut, once he is out of the spaceship, is zero.

• The second law of motion refers to the general situation when there is a net external force acting on the body. It relates the net external force to the acceleration of the body.

`"Momentum"`

• Momentum, `vecp` of a body is defined to be the product of its mass m and velocity `vecv`, and is denoted by `vecp` :

`vecp = m vecv`

• Momentum is a vector quantity.

• If two stones, one light and the other heavy, are dropped from the top of a building, a person on the ground will find it easier to catch the light stone than the heavy stone. The mass of a body is thus an important parameter that determines the effect of force on its motion.

`S"econd Law of Motion"`

The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.

If under the action of a force `vecF` for time interval Δt, the velocity of a body of mass m changes from `vecv` to `vecv + Δvecv` i.e. its initial momentum `vecp = m vecv` changes by `Δvecp =mΔvecv`.

According to the Second Law, `vecF prop (Deltavecp)/(Deltat)` or `vecF=k (Deltavecp)/(Deltat)`

Taking the limit Δt → 0

`vecF=k lim_(Deltat->0) (Deltavecp)/(Deltat)`

`vecF=k (dvecp)/(dt)`

For a body of fixed mass m,

`(dvecp)/(dt) =d/(dt) (mvecv)=m(dvecv)/(dt)=mveca`

`=> vecF=kmveca`

Here `k=1` then second law, `vecF=(dvecp)/(dt)=mveca`

`"1 Newton"`

In SI unit force is one that causes an acceleration of `1 m s^(-2)` to a mass of `1 kg`. This unit is known as newton : `1 N = 1 kg m s^(-2)`.

`"Some important points about the second law :"`

• In the second law, `vecF = 0` implies `veca = 0`. The second Law is obviously consistent with the first law.

• The second law of motion is a vector law. It is equivalent to three equations, one for each component of the vectors :

`F_x =(dp_x)/(dt)=ma_x`

`F_y =(dp_y)/(dt)=ma_y`

`F_z =(dp_z)/(dt)=ma_z`

• Any internal forces in the system are not to be included in F.

`"Momentum"`

• Momentum, `vecp` of a body is defined to be the product of its mass m and velocity `vecv`, and is denoted by `vecp` :

`vecp = m vecv`

• Momentum is a vector quantity.

• If two stones, one light and the other heavy, are dropped from the top of a building, a person on the ground will find it easier to catch the light stone than the heavy stone. The mass of a body is thus an important parameter that determines the effect of force on its motion.

`S"econd Law of Motion"`

The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.

If under the action of a force `vecF` for time interval Δt, the velocity of a body of mass m changes from `vecv` to `vecv + Δvecv` i.e. its initial momentum `vecp = m vecv` changes by `Δvecp =mΔvecv`.

According to the Second Law, `vecF prop (Deltavecp)/(Deltat)` or `vecF=k (Deltavecp)/(Deltat)`

Taking the limit Δt → 0

`vecF=k lim_(Deltat->0) (Deltavecp)/(Deltat)`

`vecF=k (dvecp)/(dt)`

For a body of fixed mass m,

`(dvecp)/(dt) =d/(dt) (mvecv)=m(dvecv)/(dt)=mveca`

`=> vecF=kmveca`

Here `k=1` then second law, `vecF=(dvecp)/(dt)=mveca`

`"1 Newton"`

In SI unit force is one that causes an acceleration of `1 m s^(-2)` to a mass of `1 kg`. This unit is known as newton : `1 N = 1 kg m s^(-2)`.

`"Some important points about the second law :"`

• In the second law, `vecF = 0` implies `veca = 0`. The second Law is obviously consistent with the first law.

• The second law of motion is a vector law. It is equivalent to three equations, one for each component of the vectors :

`F_x =(dp_x)/(dt)=ma_x`

`F_y =(dp_y)/(dt)=ma_y`

`F_z =(dp_z)/(dt)=ma_z`

• Any internal forces in the system are not to be included in F.

Q 1782523437

A bullet of mass 0.04 kg moving with a speed of

90 m/ s enters a heavy wooden block and is stopped

after a distance of 60 cm. What is the average

resistive force exerted by the block on the bullet?

90 m/ s enters a heavy wooden block and is stopped

after a distance of 60 cm. What is the average

resistive force exerted by the block on the bullet?

The retardation ‘a’ of the bullet (assumed constant) is given by

`color{green} { a = (-u^2)/(2s)}`

`= -(90xx90)/(2xx0.6)`

` = - 6750` `m//s^2`

The retarding force, by the second law of motion, is

`= 0.04 kg × 6750 m s^-2 = 270 N`

The actual resistive force, and therefore, retardation of the bullet may not be uniform. The answer therefore, only indicates the average resistive force.

Q 1712623530

The motion of a particle of mass m is descl'ibed by

`y = ut + 1/2 g t^2` Find the force acting on the particle.

`y = ut + 1/2 g t^2` Find the force acting on the particle.

`color{purple} {F = ma}` from the equation of motion a= g, so force

`color{green} {F = mg, y = ut + 1/2 g t^2}`

`color{orange} {v = (dy)/(dt) = u + g t}`

` color{red} {a = (dv)/(dt) = g}`

Then the force is given by Eq. (5.5)

`color{purple} {F = ma = mg}`

Thus the given equation describes the motion of a particle under acceleration due to gravity and y is the position coordinate in the direction of g.

Q 1762623535

A batsman hits back at ball straight in the direction

of the bowler without changing its initial speed of

12 m/s. lf the mass of the ball is 0.15 kg. Find the

impulse imparted to the ball. (assume linear motion

of the ball)

of the bowler without changing its initial speed of

12 m/s. lf the mass of the ball is 0.15 kg. Find the

impulse imparted to the ball. (assume linear motion

of the ball)

Change in momentum

`= 0.15 × 12–(–0.15×12)`

`= 3.6 N s,`

Impulse = 3.6 N s,

in the direction from the batsman to the bowler.

This is an example where the force on the ball by the batsman and the time of contact of the ball and the bat are difficult to know, but the impulse is readily calculated.

• When a large force acts for a very short duration producing a finite change in momentum of the body then change in momentum is known as impulse.

• The product of force and time, which is the change in momentum of the body remains a measurable quantity. This product is called impulse:

`"Impulse = Force × time duration = Change in momentum"`

• A large force acting for a short time to produce a finite change in momentum is called an impulsive force.

• The product of force and time, which is the change in momentum of the body remains a measurable quantity. This product is called impulse:

`"Impulse = Force × time duration = Change in momentum"`

• A large force acting for a short time to produce a finite change in momentum is called an impulsive force.