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. Its SI unit is Kgm/s.

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 M1 and M2 having the same direction with velocities u1 and u2  respectively (u1 > u2). After some time, ball A collides against ball B, The time of collision is Dt. After the collision, the ball A and B move with Velocities V1 and V2 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

FAB = (M2V2 – M2U2) / Dt

Similarly, Force exerted by ball B on A (FBA)

= (M1V1 – M1U1) / Dt

From Newton’s 3rd Law,

Action = (-ve) Reaction

FAB = -FBA

Or, (M2V2 – M22) / Dt  = - [(M1V1 – M11) / Dt]

Or, M2V2 – M22 = - M1V1 + M11

Therefore,  M1U1 + M2U2 = M1V1 + MV2

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 (U2 = 0)

Then, M1U1 = M1V1 + MV2

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

i.e., V1 = V2 = V

Then,

MU1 + M2U2 = M1V + M2V

Therefore, MU1 + M2U2 = V (M1 + M2)

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 (M1) = 0.1 kg

Velocity (U1) = 6 m/s

Bal 2,

Mass (M2) = 0.2 kg

Velocity (U2) = 0m/s

Let their common velocity be V

We know that,

From the conservation of linear momentum,

M1U1 + M2U2 = M1V1 + MV2

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/s2. Calculate the reaction of the floor when a man of mass 50 Kg standing on the lift.

Soln:

Here,     Acceleration = 1 m/s2

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.

Soln:

Here,

From the linear momentum’s conservation

We know that,

Shell before collision = shell after collision

M X U = M1V1 + M2V2

Or, 30 X 48 = 12 X V1 + 18X 0

Or, 1440 = 12V1

Or, V1 = 1440/12

Therefore, V1 = 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.

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