ISNS 3371 - Phenomena of Nature A Rifle and a Bullet When a bullet is fired from a rifle, the rifle recoils due to the interaction between the bullet and.

ISNS 3371 - Phenomena of Nature

A Rifle and a Bullet

When a bullet is fired from a rifle, the rifle recoils due to the interaction between the bullet and the rifle.

The force the rifle exerts on the bullet is equal and opposite to the force the bullet exerts on the rifle.

But the acceleration of the bullet is much larger that the acceleration of the rifle - due to Newton’s 2nd law: a = F/m

The acceleration due to a force is inversely proportional to the mass.

The force on the rifle and the bullet is the same but the mass of the rifle is much larger than the the mass of the bullet so the acceleration of the rifle is much less than the acceleration of the bullet.


Tension

Consider a block being pulled by a rope. The person doing the pulling at one end of the rope is not in contact with the block, and cannot exert a direct force on the block. Rather a force is exerted on the rope, which transmits that force to the block. The force experienced by the block from the rope is called the stretching force, commonly referred to as tension.

Tension is actually not a force - tension transmits the stretching force. A force always has a direction - the tension in a string or rope must follow the rope! The tension may have to extend around corners, over and under pulleys, etc. So, tension transmits a force through a string or rope, but tension is not a force. Tension doesn't work exactly the way force does.


Suppose you hang a 5 Newton weight from a string, and hold the other end of the string in your hand. If the weight (and the string and your hand) is at rest, then the weight exerts a 5 N downward force on the lower end of the string, and you exert a 5 N upward force on the upper end of the string. What is the stretching force/tension in the string? It is possible to build very plausible arguments that the tension in the string is 10 N, or that it is 0 N, or that it is 5 N - but what is it, really, and why?

Remember - tension transmits the force. It would be the same as if you were holding the weight in your hand - the force on your hand would be 5 N. Therefore the stretching force/tension is 5 N.

In a tug-of-war, the tension in the rope is produced by the people pulling on opposite ends of the rope. The forces at either end of the rope are equal and opposite. What is the tension in the rope?

What happens if a 200 lb man wearing socks and a 100 lb girl wearing rubber-soled shoes have a tug-of-war? Who wins?


Momentum

Momentum is mass times velocity, a vector quantity:Mom=mv

The more massive an object, the greater its momentum.

The greater the velocity of an object, the larger its momentum.

The momentum of an object is changed by applying a force:

- the larger the applied force, the greater the change in momentum.

- the longer the force is applied, the greater the change in momentum

ISNS 3371 - Phenomena of Nature Impulse

Impulse of a force is the force times the time over which the force acts on a body.

I = F x ∆T

∆ means a change in a quantity - ∆T is the time over which the force is acting.

From Newton’s second law:

Therefore, an Impulse produces a change in momentum of a body.

€

F = ma = mΔv

Δt

I = FΔt = mΔv

ΔtΔt

I = mΔv


Process of minimizing an impact force - approached from the definition of the impulse of force: If an impact stops a moving object, then the change in momentum is a fixed quantity, and extending the time of the collision will decrease the impact force by the same factor.

This principle is applied in many common-sense situations:

• If you jump to the ground from any height, you bend your knees upon impact, extending the time of collision and lessening the impact force.

• A boxer moves away from a punch, extending the time of impact and lessening the force.

• Automobiles are made to collapse upon impact, extending the time of collision and lessening the impact force.

• If you drop a glass on hard floor - it breaks. If you drop it on a soft carpet, the impact time is extended as the glass sinks into the carpet - impact force reduced - glass doesn’t break.


Conservation of Momentum

Law of Conservation of MomentumThe total momentum of an isolated system is conserved, I.e., it remains constant.

An outside or external force is required to change the momentum of an isolated system.

The Law of Conservation of Momentum is an alternate way of stating Newton’s laws:

1. An object’s momentum will not change if left alone2. A force can change an object’s momentum, but…3. Another equal and opposite force simultaneously changes some other object’s momentum by same amount

ISNS 3371 - Phenomena of NatureCollisions

net momentum before collision = net momentum after collision

In a collision, momentum is conserved because the forces acting are internal forces - momentum is simply redistributed.

ISNS 3371 - Phenomena of NatureElastic Collisions

An elastic collision is one in which the objects collide without generating heat or being permanently deformed. The objects do not stick together - they “bounce”.

Given two masses, m1 and m2 at initial velocities v1 and v2

After they collide, they have velocities V1 and V2

Conservation of momentum says that

€

m1v1 + m2v2 = m1V1 + m2V2

€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2

Solving for V1 and V2 (and using conservation of energy) gives


€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2

Let v2 = 0 and m1 = 475 gr and m2 = 266 gr

€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2

€

V1 =475 − 266

475 + 266v1 =

209

741v1

V2 =2(475)

475 + 266v1 =

950

741v1

€

m1v1 = m1V1 + m2V2

€

475v1 = 475(209

741)v1 + 266(

950

741)v1 = 475v1


€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2


€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2

€

V1 =475 − 266

475 + 266v1 =

209

741v1

V2 =2(475)

475 + 266v1 =

950

741v1

€

m1v1 = m1V1 + m2V2

€

475v1 = 475(209

741)v1 + 266(

950

741)v1 = 475v1

m1(the heavier car) is still moving after the collision, but slower.m2(the lighter car) is moving after the collision with a velocity greater than the velocity of m1 before the collision.

Momentum is conserved

A heavy car collides with a stationary lighter car


€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2


€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2

€

V1 =266 − 475

475 + 266v1 =

−209

741v1

V2 =2(266)

475 + 266v1 =

532

741v1

€

m1v1 = m1V1 + m2V2

€

266v1 = 266(−209

741)v1 + 475(

532

741)v1 = 266v1

m1(the lighter car) is still moving after the collision, but in the opposite direction.m2(the heavier car) is moving after the collision with a velocity smaller than the velocity of m1 before the collision.


A light car collides with a stationary heavier car


€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2

m1 = m2

€

V1 =m1 −m2

m1 + m2

v1 +2m2

m1 + m2

v2

V2 =2m1

m1 + m2

v1 +m2 −m1

m1 + m2

v2

€

m1v1 + m2v2 = m1V1 + m2V2 = m1v2 + m2v1

m1 and m2 simply switch velocities - it doesn’t matter whether they are going in the same or opposite directions.


Two moving cars with the same mass collide

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V1 =2m2

m1 + m2

v2 = v2

V2 =2m1

m1 + m2

v1 = v1


Elastic Collisions in 2 Dimensions - Pool

Remember: Momentum is a vector quantity - so the vector sum of the two balls’ momentum must equal the momentum of the que ball (the red ball) before collision.

Note: the angle that the que ball and the object ball make after collision is always a right angle (we will show this later).


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q = que ball

o = object ball

mqvq1 = mqvq2 cos(α 2) + movo cos(α 1)

mq = moso

vq1 = vq2 cos(α 2) + vo cos(α 1)


Inelestic Collisions

In an inelastic collision, the objects stick together after collision. Again, momentum is conserved:

€

m1v1 + m2v2 = m1V1 + m2V2

V1 = V2 because the objects are moving together and:

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m1v1 + m2v2 = (m1 + m2)V

€

V =m1v1 + m2v2

m1 + m2


€

V =m1v1 + m2v2

m1 + m2

€

V =m1v1 + m2v2

m1 + m2

=m1

m1 + m2

v1 =1

2v1

€

V =m1v1 −m2v1

m1 + m2

= 0

Two cars of the same mass and velocities equal but in the opposite direction: v2 = -v1

Velocity after the collision is 0

Two cars of the same mass, one moving and the other stationary: v2 = 0

Velocity after collision is

1/2 velocity of m1 before collision


Angular Momentum

Momentum associated with rotational or orbital motionangular momentum = mass x velocity x radius


Torque and Conservation of Angular Momentum

Conservation of angular momentum - like conservation of momentum -in the absence of a net torque (twisting force), the total angular momentum of a system remains constant

Torque - twisting force


A spinning skater speeds up as she brings her arms in and slows down as she spreads her arms because of conservation of angular momentum

ISNS 3371 - Phenomena of Nature A Rifle and a Bullet When a bullet is fired from a rifle, the rifle recoils due to the interaction between the bullet and.

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