**DYNAMICS**

**Dynamic**

It is the branch of physics which deals
with the motion of objects and factors affecting them i.e., Force, Mass,
Momentum, Energy, etc.

**Force**

Force is an external agent which changes or
tends to change the state of the rest of a body or its uniform motion. Force
applied to a body can change the following things:-

1.
A
force can change the speed of motion.

2.
A
force can change the direction of motion.

3.
A
force can change the shape and size of the body.

**Inertia**

It is the property of an object to continue its
original state. It cannot itself change its state of rest or its uniform
motion.

**Types of Inertia**

1. Inertia of rest

2. Inertia of motion

3. Inertia of direction

#### 1. Inertia of rest

In the property of the body to remain in its state of
rest. Example:

i) when we shake an apple
tree, the apple falls down, It is because we shake the tree, the apples tend to be at rest due to inertia whereas the branches are in motion. That is why the
apples get detached from the branches.

ii) A man standing on a
bus falls backwards when the bus suddenly starts from rest. This is due to the inertia
of the rest of the passengers.

iii) The dust particles
are removed from a carpet by beating it because due to the particle's inertia of
the rest the dust particles try to remain at rest.

#### 2. Inertia for motion

It is the property of a body to remain in its uniform
motion. Example:

i) A man standing on the
bus falls forward when the bus stops suddenly.

ii) An athlete runs
before he takes his jump.

#### 3. Inertia of direction

The property of a body to remain in its direction of motion
is called inertia of direction. Example:-

i) When a bus takes a turn, the passengers
sitting in the bus tend to move tangentially.

**Linear momentum:**

The product of the mass and velocity of a body is called
linear momentum, It is denoted by Vector P. It is a Vector quantity.
i.e., Vectr P = M x Vector V

**Force **

The rate of change of linear momentum is called force.
i.e., F = dp/dt

**Principle of conservation of linear momentum**

Statement:

If no external force acts on the system then the total
momentum of the system remains constant. i.e.,
Total linear momentum before collision = Total linear momentum after the collision.

Proof:

Suppose two spheres A and B of masses M_{1} and
M_{2} having the same direction with velocities u_{1} and u_{2} respectively (u_{1} > u_{2}).
After some time, ball A collides against ball B, The time of collision is **D****t**. After the collision, the
ball A and B move with Velocities V_{1} and V_{2}
respectively.

If the force exerted by ball A on ball B is the action,
then the force exerted by ball B on ball A is the reaction.

From Newton’s second law,

Force exerted by ball A
on B

= Rate of change of
momentum of B

= Change in the momentum of B
/ Time taken

F_{AB} = (M_{2}V_{2}
– M_{2}U_{2}) / Dt

Similarly, Force exerted
by ball B on A (F_{BA})

= (M_{1}V_{1} – M_{1}U_{1}) / Dt

From Newton’s 3^{rd}
Law,

**Action = (-ve) Reaction**

**F _{AB} = -F_{BA}**

Or, (M_{2}V_{2}
– M_{2}U_{2}) / Dt = - [(M_{1}V_{1} – M_{1}U_{1})
/ Dt]

Or, M_{2}V_{2}
– M_{2}U_{2} = - M_{1}V_{1} + M_{1}U_{1}

Therefore, M_{1}U_{1} + M_{2}U_{2}
= M_{1}V_{1} + M_{2}V_{2}

i.e., total linear
momentum of BC = total linear momentum of AC

This proves the principle
of conservation of linear momentum.

Special cases (i) :- If
body B is in rest (U_{2} = 0)

Then, M_{1}U_{1 = }M_{1}V_{1}
+ M_{2}V_{2}

(ii):- If after the collision both balls
move with the same velocity (V)

i.e., V_{1} = V_{2}
= V

Then,

M_{1}U_{1}
+ M_{2}U_{2} = M_{1}V + M_{2}V

Therefore, M_{1}U_{1}
+ M_{2}U_{2} = V (M_{1} + M_{2})

**Q. A ball of mass 0.1 kg
moving with a velocity of 6 m/s collides directly with another ball of mass 0.2
kg at rest, calculate their common velocity, if both balls move together.**

** **Here, There are two
balls,

Ball 1,

Mass (M_{1}) = 0.1 kg

Velocity (U_{1}) = 6 m/s

Bal 2,

Mass (M_{2}) = 0.2 kg

Velocity (U_{2}) = 0m/s

Let their common velocity
be V

We know that,

From the conservation of linear momentum,

M_{1}U_{1} + M_{2}U_{2} = M_{1}V_{1}
+ M_{2}V_{2}

Or, 0.1 X 6 + 0.2 X 0 = 0.1 X V + 0.1 X V

Or, 0.6 = (0.1 +0.2 ) V

Or, 0.6 = 0.3V

Or, V = 0.6/0.3

Therefore, V = 2 m/s

The common velocity of
two balls is 2 m/s.

**Solve!!**

**Q. Ball A of mass 0.1 kg
moving with a velocity of 6 m/s collides directly with ball B of mass 0.2 kg at
rest. Calculate the common velocity in both balls moving off together. If A had
rebounded with a velocity of 2 m/s in the opposite direction after collision. What
would be the new velocity of B?**

**Apparent Weight**

The
reaction of the machine on power is called apparent weight.

Case
1:- When the lift is moving upward:-

Net force = R – W

F
= R – W

Or, R = F + W

Or, R = Ma + Mg

Therefore, Apparent weight increases as the lift move upward.

Case 2:- When the lift is moving downward

Net force = W – R

F
= W – R

R
= Mg – Ma

Therefore, Apparent weight decreases as the lift move
downwards.

Note: During freefall (a
= g)

R = Mg – Mg

Therefore, R = 0

Case 3:- When the lift
is at rest,

Net force = 0

R = W

R = Mg

**Q. A list moves up with a
constant acceleration of 1 m/s ^{2}. Calculate the reaction of the
floor when a man of mass 50 Kg standing on the lift.**

Soln:

Here, Acceleration = 1 m/s^{2}

Mass of man = 50 kg

Acceleration due to
gravity (g) = 10 m/s

Reaction =?

We have, When lift moves
upward,

R = Ma + Mg

= 50 X 1 + 50 X 10

= 50 + 500

= 550 N

Hence, the reaction of
man is 550 N.

**Q. A 30 Kg shell is
flying at 48 m/s. When it explodes, Its one part of 18 kg stops while the
remaining part flies on. Find the velocity of the remaining part.**

Here,

From the linear
momentum’s conservation

We know that,

Shell before collision =
shell after collision

M X U = M_{1}V_{1}
+ M_{2}V_{2}

Or, 30 X 48 = 12 X V_{1}
+ 18X 0

Or, 1440 = 12V_{1}

Or, V_{1} =
1440/12

Therefore, V_{1}
= 120 m/s

**Moment of Force or Torque**

** **The turning effect of force about an axis is called
torque. It is defined as the product of the force and the perpendicular
distance of the line of action from the axis of rotation.

i.e., Moment of force or torque = Force X perpendicular
distance

or, t (tau) = F X ^^{r} distance

The unit of torque is Newton – Metre (Nm)

**Principle of moment**

** **The principle of moment states that “for the object in equilibrium,
the sum of the clockwise moment about any point provided by the force acting on
the object equals to the sum of anti-clockwise

Moment of any point.

Sum of clockwise moment = Sum of anti-clockwise moment

**Impulse**

It
is defined as the product of the average force applied and the time for which the
force is applied.

i.e., Impulse (I) = Force X Time of impact

or, I = F X t

Unit of Impulse = Newton-Second (Ns)

**Application of Impulse**

1. A person falling on a cemented floor gets more injury
than one falling on sand.

è Because the rate of change of momentum is less because of which, a smaller stopping
force acts on and he doesn’t get hurt. Thus sand being soft reduces the man’s
momentum more gently.

2. A cricketer lowers his hand while catching a fast-moving ball.

è Because the time of impact increases the rate of change of linear momentum
decreases and the forces acting on the hands are reduced. However, the change
in linear momentum or impulse is the same in both cases.

3. Chinaware is wrapped in straw or paper while packing.

è Because in the event of a fall, the impact will take a longer time to reach the
chinaware through the straw of paper; hence, the average force exerted on the
chinaware is small and the chance of it breaking will be reduced.

4. It is easier to catch a tennis ball as compared to a cricket ball moving with the same velocity.

è Because the mass of a cricket ball is more than that of a tennis ball and hence
cricket ball has more momentum. Therefore, it is easier to stop a tennis ball
than a cricket ball moving at the same speed.