**Force**

Force is defined as an external agent that can change or tries to change the state of rest or uniform motion of an object. It’s unit is Newton (N) in S.I. and Dyne in C.G.S. It is a vector quantity.

1N = 10^{5} dyne

**Inertia **

The tendency of an object to remain in its own state of rest or uniform motion is called inertia.

**Types of inertia **

(i) Inertia of rest

(ii) Inertia of motion

(iii) Directional inertia

**Newton’s first law of motion **

Everybody in the universe continues in its own state of rest or uniform motion unless an external force acts upon it.

i.e. for F_{external} = 0, u = v

**Q. Newton’s first law of motion gives the definition of force and inertia. Justify. **

Solution:

Newton’s first law of motion describes force in a qualitative way. According to the first law of motion, an object cannot change its state of rest or uniform motion without an external force. Thus, it can be concluded that force is an external agent that causes a body to change its state (rest or uniform motion). Thus, Newton’s first law of motion defines force and inertia.

**Momentum (Linear momentum,$\overrightarrow{P}$) **

It is defined as the total motion contained in a body. It is denoted by$\overrightarrow{P}$ and mathematically given by

$\overrightarrow{P}$ = mass × velocity

= m × v

It’s unit is kgms^{–1}.

**Newton’s second law of motion **

It states that the rate of change of momentum of a body is directly proportional to the force applied on it,

i.e. $\frac{\Delta p}{\Delta t}$$\propto $ F

Let us consider an object of mass ‘m’ moving with velocity ‘u’ at position A, and ‘a’ be its uniform acceleration. After time ‘t’, let the velocity of the object become ‘v’ at position B.

Now, initial momentum = m.u

final momentum = m.v

change in momentum = mv – mu

Rate of change in momentum = $\frac{mv-mu}{t}$

According to Newton’s second law of motion, rate of change of momentum is directly proportional to the force applied on it,

i.e. $\frac{mv-mu}{t}$ $\propto $ F

or, $\frac{m(v-u)}{t}$ $\propto $ F

or, ma $\propto $ F

or, ma = kF, [where K is proportionality constant, for K = 1]

$\therefore F=ma$

Thus, Newton’s second law of motion gives the quantitative definition of force.

By knowing the values of ‘m’ and ‘a’, force can be calculated.

*Q. Newton’s second law of motion gives the measurement of force. Justify.*

**Impulse (I)**

It is defined as the force acting upon an object for a very short time that changes the momentum. The short time for which the objects remain in contact is known as time of impact. Impulse is denoted by ‘I’ and mathematically given by

I = F × $\Delta $t

= ma × $\Delta $t

= $\frac{m(v-u)}{\Delta t}$ × $\Delta $t

Or, I = mv – mu

$\therefore $I = P_{f }– P_{i}

Thus, numerically impulse is equal to change in momentum.

**Newton’s third law of motion **

It states that to every action, there is an equal and opposite reaction.

Let us consider two objects A and B collide with each other. Let F_{BA} and F_{AB }be the forces exerted by A on B and by B on A respectively. Then, according to Newton’s third law of motion.

F_{BA} = – F_{AB}

**(i) Derivation of first law of motion from second law **

From Newton’s second law of motion,

F = $\frac{dp}{dt}$ = $\frac{m(v-u)}{t}$

In the absence of external force, i.e. F = 0,

We have,

Or, 0 = $\frac{m(v-u)}{t}$

or, v – u = 0

$\therefore $ v = u

i.e. an object continues in its own state (rest or uniform motion, u = v) in the absence of external force. This is Newton’s first law of motion.

**(ii) Derivation of third law of motion from second law **

From Newton’s second law of motion,

F = $\frac{dp}{dt}$ = $\frac{\Delta p}{\Delta t}$ ………. (i)

Let us consider two bodies A and B of masses ‘m_{1}‘ and ‘m_{2}‘ are moving with initial velocities ‘u_{1}‘ and ‘u_{2}‘ respectively such that u_{1} is greater than u_{2}. They collide after some time and move separately with final velocities ‘v_{1}‘ and ‘v_{2}‘ respectively. During the collision, let F_{BA} be the force exerted by A on B and F_{AB} be the force exerted by B on A respectively.

From Newton’s second law of motion,

F_{AB} = $\frac{d{{P}_{A}}}{dt}$ = $\frac{\Delta {{P}_{A}}}{t}$

$\therefore $ $\Delta $P_{A} = F_{AB} × t ……….. (ii)

And, F_{BA} = $\frac{d{{P}_{B}}}{dt}$ = $\frac{\Delta {{P}_{B}}}{t}$

$\therefore $ $\Delta $P_{B} = F_{BA} × t ………… (iii)

$\therefore $ Total change in momentum of A and B, $\Delta $P = $\Delta $P_{A} + $\Delta $P_{B}

In the absence of external force i.e. F_{ext} = 0

then from equation (i) we have,

$\Delta P$ = 0

$\therefore $ $\Delta $P_{A} + $\Delta $P_{B }= 0

From (i), (ii) and (iii)

F_{AB }× t + F_{BA }× t = 0

Or, t (F_{AB} + F_{BA}) = 0

$\therefore $F_{AB }= – F_{BA}

i.e. two forces F_{AB} and F_{BA} are equal in magnitude and opposite in direction. This is Newton’s third law of motion. Since *Newton’s first and third laws can be derived from the second law of motion. So Newton’s second law is the real law*

*Q. Newton’s second law is the real law. Explain.Or*

*Q. Derive of Newton’s first and third law from the second law of motion. *

**Principle of Conservation of Linear Momentum:**

Statement:

It states that in the absence of external force, the total linear momentum of an isolated system remains constant.

OR

In collision, the sum of linear momentum before collision is equal to the sum of linear momentum after collision in the absence of external force.

Proof:

Let us consider two bodies A and B of masses ‘m_{1}‘ and ‘m_{2}‘ are moving with initial velocities ‘u_{1}‘ and ‘u_{2}‘ respectively such that u_{1} is greater than u_{2}. They collide after some time and move separately with final velocities ‘v_{1}‘ and ‘v_{2}‘ respectively. During the collision, let F_{BA} be the force exerted by A on B and F_{AB} be the force exerted by B on A respectively.

Now,

initial momentum of A = m_{1}u_{1}

initial momentum of B = m_{2}u_{2}

final momentum of A = m_{1}v_{1}

final momentum of B = m_{2}v_{2 }

change in momentum of A = m_{1}v_{1} – m_{1}u_{1}

change in momentum of B = m_{2}v_{2} – m_{2}u_{2}

_{}Then,

Rate of change of momentum of A = $\frac{{{m}_{1}}{{v}_{1}}-{{m}_{1}}{{u}_{1}}}{t}$

Rate of change of momentum of B = $\frac{{{m}_{2}}{{v}_{2}}-m{}_{2}{{u}_{2}}}{t}$

According to Newton’s second law of motion.

Force exerted by A on B = rate of change of momentum of B.

i.e. F_{BA} = $\frac{{{m}_{2}}{{v}_{2}}-m{}_{2}{{u}_{2}}}{t}$……… (i)

Force exerted by B on A = rate of change of momentum of A

i.e. F_{AB} = $\frac{{{m}_{1}}{{v}_{1}}-{{m}_{1}}{{u}_{1}}}{t}$……… (ii)

According to Newton’s third law of motion,

F_{BA} = – F_{AB}

Or, $\frac{{{m}_{2}}{{v}_{2}}-m{}_{2}{{u}_{2}}}{t}$ = $\frac{-(m{}_{1}{{v}_{1}}-{{m}_{1}}{{u}_{1}})}{t}$

or, – m_{1}v_{1} + m_{2}u_{2} = m_{2}v_{2} – m_{1}u_{1}

$\therefore $ m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2},

i.e. the sum of linear momentum before collision is equal to the sum of momentum after collision which is the principle of conservation of linear momentum.

**Some applications of conservation of linear momentum **

**Motion of gun and bullet**

Let us consider a gun and a bullet of masses m_{1 }and m_{2} respectively are at rest before firing. After firing, let the gun and the bullet move with velocities v_{1} and v_{2}. From conservation of linear momentum,

m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}

Since, u_{1} = u_{2} = 0

m_{1}v_{1} + m_{2}v_{2} = 0

$\therefore {{v}_{1}}=\frac{-{{m}_{2}}{{v}_{2}}}{{{m}_{1}}}$

i.e. the gun and the bullet move in opposite directions, which is explained by the minus sign.

**Common velocity**

When two bodies collide with each other and if they combine after collision and move with the same velocity, that velocity is known as common velocity.

The collision of two bodies and their common velocity are shown above.

From the conservation of linear momentum m_{1}u_{1} + m_{2}u_{2 }= m_{1}v_{1} + m_{2}v_{2}

After collision, if they move with the same velocity, v_{1 }= v_{2} = v_{c} (common velocity)

m_{1}u_{1} + m_{2}u_{2} = m_{1}v + m_{2}v

or, m_{1}u_{1} + m_{2}u_{2} = v_{c}(m_{1} + m_{2})

$\therefore {{v}_{c}}=\frac{{{m}_{1}}{{u}_{1}}+{{m}_{2}}{{v}_{2}}}{{{m}_{1}}+{{m}_{2}}}$

**Free body diagram:**

The representation of bodies (mass) by points and various forces acting on them is called a free body diagram.

**Types of forces to be discussed in free body diagram **

(i) Weight, w = mg

(ii) Reaction, R

(iii) Tension, T

(iv) Upthrust, U

(v) F_{f} is frictional force

**Equation of Force**

**Applied force – Opposing force = Resultant force **

**i.e. F**_{app}** – F**_{opp}** = ma**

**(a)**** ****Motion of lift **

**(i) When the lift is moving upward**

R – mg = ma

$\therefore $ Effective weight, R = mg + ma (increases)

**(ii)**** ****When the lift is moving downward **

mg – R = ma

$\therefore $ Effective weight, R = mg – ma (decreases)

**(iii)**** ****When the lift is moving downward with a = g **

Effective wt. mg – ma = m(g – a) = 0 (weightlessness/ free fall)

**(iv)**** ****Downward with a > g **

Effective wt. = mg – ma = – ve

The body will strike the ceiling of the lift.

**(b) Motion on frictionless pulley **

**(i)**** ****For upward motion **

T – m_{1}g = m_{1}a

**(ii)**** ****For downward motion **

m_{2}g – T = m_{2}a

**Frictional force (F**_{f}**):**

_{f}

It is defined as the opposing force that comes in play between the contact surfaces of two bodies when one moves, slides, rolls or tends to do so.

**Causes of Friction **

**Classical concept**

There are a large number of irregular projections on the surfaces of bodies, when one body is placed in contact with another and external force is applied, then these irregular projections begin to interlock and oppose the motion of the body. This is called frictional force.

**Molecular concept**

With increase in smoothness of the surfaces of two bodies in contact the area of contact also increases which increases the intermolecular force of attraction. Hence, the frictional force also increases. This is called the molecular concept of friction or cold-welding.

**Types of Friction **

**(i) Static Friction (F**_{FS}**)**

_{FS}

It is defined as the frictional force which comes in play between the surfaces of two bodies, when one is in the state of rest over the surface of another and external force is applied. Its value is always equal to applied force.

i.e. F_{FS} = F_{a}

**(ii) Limiting friction (F**_{FL}**)**

_{FL}

It is the maximum value of static frictional force or minimum value of force required just to move the object.

**(iii) Kinetic friction (F**_{FK}**)**

_{FK}

It is defined as the frictional force which comes in play between the surfaces of two bodies, when one is in the state of motion over the surface of another body. It is the force required to keep a body moving with uniform velocity.

**(iv) Rolling friction (F**_{FR}**)**

_{FR}

It is defined as the frictional force which comes in play between the surfaces of two bodies, when one body rolls over the surface of another body.

Note:

Static friction > Kinetic friction > Rolling friction

**Coefficient of Friction ($\mu $)**

The frictional force is directly proportional to normal reaction, i.e.

F_{f} $\propto $ R

F_{f} = $\mu $R

$\mu $ = $\frac{{{F}_{f}}}{R}$

where, $\mu $ is proportionality constant known as coefficient of friction between the two surfaces.

It is also defined as the ratio of frictional force to normal reaction. It has no unit.

**Laws of friction:**

The laws of friction are:

(i) The friction force opposes the motion of the body and acts tangentially to the surface in contact.

(ii) The frictional force is directly proportional to the normal reaction

i.e. F_{f} $\propto $ R

or, F_{f} = $\mu $R

or, $\mu $ = $\frac{{{F}_{f}}}{R}$, where $\mu $ is the proportionality constant known as coefficient of friction between two surfaces.

(iii) The frictional force depends upon roughness or smoothness of the surface.

(iv) It is independent of the area of surface in contact.

Experimental verification

Let us consider a wooden block placed on a horizontal surface. It is attached to a string which passes over a frictionless pulley carrying a scale pan at free end.

Small weights are added gradually on the pan till the wooden block just starts to slide. This shows the frictional force opposes the motion of the body and acts tangentially to the surface in contact verifying the first law.

When the block just starts to slide, the total weight added on the scale pan is equal to the limiting frictional force. The normal reaction (R) is equal to the weight of wooden black and the ratio $\frac{{{F}_{f}}}{R}$is calculated. The experiment is repeated by placing different masses on wooden blocks and forces of limiting friction are found each time. The rates are found to be constant. This verifies the second law.

The wooden block is replaced by blocks of different materials of the same weight. It is found that frictional force is different in each case. This verifies the third law.

If the wooden block is placed on its different sides with different areas, the same limiting frictional force is found which verifies the fourth law.

**Q. What are the laws of friction? How are they verified? **

**Angle of Friction ($\theta $):**

It is defined as the angle made by the resultant of frictional force (F_{f}) and normal reaction (R) with normal reaction (R).

Let us consider an object of mass ‘m’ is placed on a horizontal surface. A force F_{a }is applied on the object so that the object just starts to move. The coefficient of friction between the object and the surface is $\mu $. The weight ‘mg’ acts vertically downward and normal reaction acts normally upward to the surface along OA. When the object just starts to move then applied force (F_{a}) = Limiting frictional force (F_{f}). Frictional force (F_{f}) acts in the direction opposite to the direction of motion along OC. Let us complete a parallelogram OABC. According to parallelogram law of vector addition, OB = F_{R }gives the resultant of frictional force (F_{f}) and normal reaction (R) as in figure above. By the definition, $\angle $AOB = $\theta $ is the angle of friction.

Now, tan$\theta $ = $\frac{AB}{OA}$ = $\frac{OC}{OA}$

$\therefore $ tan$\theta $ = $\frac{{{F}_{f}}}{R}$

$\therefore $ $\mu $ = $\frac{{{F}_{f}}}{R}$

**$\therefore $$\mu $= tan$\theta $**…….(a)

This shows that coefficient of friction is equal to tangent of angle of friction.

**Angle of Repose ($\alpha $):**

It is defined as the angle made by an inclined plane with horizontal at which an object placed over it just starts to slide.

Let us consider an object of mass ‘m’ just slides on an inclined plane which makes an angle α with the horizon. The weight ‘mg’ acts vertically downward, normal reaction ‘R’ acts normally outward to the surface. The frictional force ‘F_{F}‘ acts in the direction opposite to that of motion as in figure. Resolving the force ‘mg’ into its constituent, we get mgsin$\alpha $ and mgcos$\alpha $ as in figure. The component mgsin$\alpha $ is balanced by frictional force F_{f},

i.e.

mgsin$\alpha $ = F_{f} …………. (i)

And the component mg cos$\alpha $ is balanced by normal reaction R,

i.e. mg cos$\alpha $ = R ……….. (ii)

Dividing (i) by (ii),

$\frac{mgsin\alpha }{mg cos\alpha }$ = $\frac{{{F}_{f}}}{R}$

or, tan$\alpha $ = $\mu $ [ $\mu $ = F_{F}/R]

$\therefore $$\mu $= tan$\alpha $…………(b)

This shows that coefficient of friction is equal to tangent of angle of repose.

From equation (a), We have,

$\mu $ = tan$\theta $

From equation (b), we have

$\mu $ = tan$\alpha $

$\therefore $ tan$\alpha $ = tan$\theta $

**$\therefore $$\alpha $** = $\theta $

i.e. angle of repose is equal to angle of friction for a given pair of surfaces.

*Q. Define angle of friction and angle of repose. Show that angle of friction is equal to angle of repose for a given pair of surfaces. *

**Normal reaction in different cases: **

**1.**** ****Horizontal surface **

R = mg

**2.**** ****Inclined plane **

R = mg cos$\theta $

**3.**** ****Lift moving upward **

R = mg + ma

R – mg = ma

**4.**** ****Lift moving downward **

mg = R = ma

R = mg – ma

**5.**** ****Case of pulling **

R = mg – Fsin$\theta $

mg = Fsin$\theta $ + R

**6.**** ****Case of pushing **

R = mg + Fsin$\theta $