We shall require a substantially new manner of thinking if mankind is to survive.

– Albert Einstein

Even though most of you will be remembering the important points in connection with Newton’s Laws of Motion, it will be better to have a glance at the following:

**(1)** Inertia is a basic property of any material body, by virtue of which it resists any change in its state of rest or of uniform motion.

**(2)** A force is required to change the state of rest or of uniform motion of a body. The resultant force acting on a body at rest or in uniform motion is zero. [Note that a body in uniform motion has uniform *velocity*].

**(3)** Newton’s second law is mathematically expressed as

**F*** _{net}* =

*m*

**a**where

**F**

*is the net (resultant) force and*

_{net}**a**is the acceleration.

This can be written in terms of momentum **p **as

**F*** _{net}* =

*d*

**p**/

*dt*

Often we write this as **F**= *d***p**/*dt*, understanding that **F** is indeed the *net* *force*.

[Remember that the mass *m* can be treated as constant only at speeds negligible compared to the speed of light].

**(4)** **Impulse** given to an object (by a force) = **F**∆*t* where **F** is the force and ∆*t* is the time for which the force acts.

If the force is not constant and it acts from the instant *t*_{1} to the instant *t*_{2}, we have

_{t}_{1}∫

^{t}^{2}

**F**

*dt*

This gives the *area under the force-time graph* between the ordinates corresponding to the times *t*_{1} and *t*_{2} (Shaded area in fig.)

Since force **F** = ∆**p**/∆*t* where ∆**p** is the change in momentum during the time ∆*t*, we can write

Impulse = **F**∆*t* = (∆**p**/∆*t*) ∆*t* = ∆**p**

Thus impulse = change of momentum

**(5) Motion in a lift**

The weight of a body of mass ‘m’ in a lift can be remembered as *m*(*g-a*)** **in all situations if you apply the proper sign to the acceleration ‘*a*’ of the lift. The acceleration due to gravity ‘*g*’ always acts vertically *downwards* and its sign may be taken as positive. The following cases can arise in this context:

(i) Lift moving down with acceleration of magnitude ‘*a*’:

In this case ‘*a*’ also is positive and the weight is *m*(*g-a*) which is less than the real weight of the body (when it is at rest).

(ii) Lift moving up with acceleration:

In this case ‘*a*’ is negative and the weight is *m*[*g-*(-*a*)]** = ***m*(*g+a*).

(iii) Lift moving down with *retardation* (going to stop while moving down):

In this case also ‘*a*’ is negative and the weight is *m*[*g*-(-*a*)] **=***m*(*g+a*) which is greater than the actual weight.

(iv) Lift moving up with *retardation* (going to stop while moving up):

** **In this case ‘*a*’ is positive and the weight is *m*(*g-a*)

(v) Lift moving up or down with *uniform velocity*: ** **

In this case ‘*a*’ is zero and the weight is *mg*.

(vi) Lift moving down with acceleration of magnitude ‘*g*’ (falling freely under gravity as is the case when the rope carrying the lift breaks):

In this case ‘*a*’ is positive and the weight is *m*(*g-g*) which is *zero*.

If you have a clear idea of the weight of a body in a lift, you will be able to use it in other similar situations as well (for instance, the motion of bodies connected by a string passing over a pulley).

[*We will discuss conservation of momentum separately in due course*].

**(6) Friction**

The force of friction, **F*** _{fric}* ≤

*μ*

**N**where

*μ*is the

*coefficient of friction*and

**N**is the

*normal reaction*(normal force).

In the adjoining figure, a body being pulled along a horizontal surface by a horizontal force **F **is shown. The frictional force **F*** _{fric}* is maximum when the body

*just begins to move*and is called limiting force of static friction (

**F**)

_{s}*so that we have*

_{max} (**F_{s}**)

*=*

_{max}*μ*

_{s}**N**. This gives the value of the

*coefficient of static friction μ*as

_{s} *μ _{s}*

_{ }= (

**F**)

_{s}

_{max}*/*

**N**

When the body slides along the surface, the friction called into play is called *kinetic friction*. The force of kinetic friction **F*** _{k}* is less than the above limiting value (

**F**)

_{s}*and the corresponding*

_{max}*coefficient of kinetic friction*

*μ*is less than

_{k}*μ*. We have

_{s} *μ _{k}* =

**F**

_{k}/**N**

If the body rolls along the surface, The friction called into play is called *rolling friction* which is much less than kinetic friction.

**Angle of friction ** ** λ **is the angle between the the normal force

**N**and the resultant

**reaction**

**S**. As shown in the figure, the resultant reaction is the resultant of the normal force

**N**and the frictional force

**F**

*. Since tan*

_{fric}

*λ*=

**F**

*/*

_{fric}**N**, it follows that

** ***μ =* tan* **λ*

A body of mass *m *placed on a ramp (inclined plane) is shown in the adjoining figure. The component *m***g** sin*θ* of the weight *m***g** of the body is the force trying to move the body down the plane. The normal reaction is the reaction (force) opposing the component *m***g*** *cos*θ* of the component of the weight of the body normal to the inclined plane. The frictional force **F*** _{fric}* is opposite to the component

*m*

**g**sin

*θ*(of the weight of the body) parallel to the plane. Note that friction is a self adjusting force up to its maximum value (

**F**)

_{s}*and if the body shown in the figure is at rest,*

_{max}**F**

*is just sufficient to balance the component*

_{fric}*m*

**g**sin

*θ*

_{ }(of the weight of the body).

If the inclination of the plane is gradually increased from a small value, the body placed on it will begin to slide down when the angle is equal to the angle of friction,* **λ*. The *angle of repose* is therefore equal to the angle of friction.

Often you may be asked to draw a ** free body diagram** (FBD), indicating the forces acting on the body. In the case of the body placed on the inclined plane, the free body diagram is as shown. The body is represented by a dot. The forces to be shown are the weight

*m*

**g**of the body, the normal force

**N**(equal to

*m*

**g**

*cos*

*θ*) exerted by the inclined surface on the body and the frictional force

**F**

_{fric}_{ }since they are the actual forces acting on the body. Don’t worry about the components

*m*

**g**sin

*θ*and

*m*

**g**cos

*θ*of the weight. The real force is the weight

*m*

**g**which we have shown already.

**We consider the components just for the convenience of explanation. The normal reaction (force) offered by the surface and the frictional force between the body and the surface are to be accommodated in addition to the weight of the body.**

[Note that if you want, you can draw the FBD showing the components *m***g** sin*θ* and *m***g** cos*θ* of the weight of the body. But in that case you will not show the weight *m***g** in the FBD].

If the inclined plane is smooth, the frictional force **F_{fric}** will be absent in the free body diagram.

If the body is held on a smooth incline by a spring fixed to the incline, the spring force **K***x* has to be shown in place of the frictional force** F_{fric}**. Here K is the spring constant and

*x*is the elongation (or compression as the case may be) of the spring.

If the body moves *down the incline* and the viscous drag force (air resistance) is significant, that too is to be shown *up the incline*.

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