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Conservation of Momentum

The principle of conservation of momentum states: When there is no net external force on a system of objects, the total momentum of the system does not change.

We say that the momentum of such a system is conserved since it does not change.

Collisions •  When two objects collide, impulse is equal and

opposite for the two objects if there is no net external force.

•  Each object has equal and opposite change in momentum.

•  The momentum of the system of 2 objects is conserved.

Before Collision

Impact

After Collision

Collisions

•  The change of momentum of the two objects in a collision is equal and opposite -- the momentum gained by one object is the amount lost by the other.

Momentum Object A

before collision

Momentum Object B

Before collision

Momentum Object A

after collision

Momentum Object B

after collision

Demo: Elastic vs. Inelastic Collisions

A sharp dart and a blunt dart will each be fired at the wooden block. The sharp dart will stick in the block, and the blunt dart will bounce off. Will they exert the same impulse on the wooden block? If not, which one will exert more impulse?

Collisions 1. Elastic collision: Objects rebound without deforming or generating heat

– Billiard balls 2. Inelastic: objects hit, but deform and/or generate heat

– Tennis ball being hit by a racket 3. Perfectly inelastic: Objects hit and stick together

•  Momentum is conserved, whether collision is elastic or inelastic, as long as there’s no net external force!

Recoil: Also explained by conservation of momentum

M x v m x V

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Example: Big fish eats smaller fish!

If the big fish has a mass 5x bigger than the small fish, what is the speed of the big fish after lunch?

Main Points:

•  Definition of Momentum •  Definition of Impulse •  Need an impulse to change momentum •  Conservation of Momentum in collisions •  Different types of collisions

Chapter 7: Work and Energy Work: The transfer of energy (something must be moved)

•  Definition of work done on an object: Work = force x distance traveled

•  W = F d •  SI Unit of work: Joule (J), 1 = 1 N!m •  d is distance along force direction

– Force acting in direction of motion: Positive work. – Force acting in opposite direction: Negative work. – Force perpendicular to motion: Zero work

Examples: Work

•  Imagine dragging a heavy box along the ground. What forces act on the box? Which forces do work? Do they do positive work, or negative work?

Work against & by Gravity

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Examples: Loading a Ship

•  3,000 kg truck is loaded onto a ship by crane that exerts upward force of 31,000 N on truck. This force is applied over a distance of 2.0 m. (a) Find work done on truck by

crane (b) Find work done on truck by gravity. (c) Find net work done on the truck.

Power

•  Power is a measure of the rate at which work is done. If work W done during time t:

P = Work time

•  SI unit: 1 J/s = 1 watt = 1 W •  Horsepower: 1 hp = 746 W

Typical Power Outputs

Human basal metabolism is around 80 W

Example: Power

•  How much power is expended when a weightlifter lifts a 500 N barbell 2.2 m in 2 seconds?

Mechanical Energy: Energy due to position or movement

•  Potential Energy: stored energy (gravitational, spring, etc.) – Gravitational PE = mgh

•  Kinetic Energy: energy associated with motion – KE = ½ mv2

•  Both have units of Joules!

Gravitational Potential Energy

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Example: Kinetic Energy

•  A person pushes a 10 kg crate so that it accelerates up to a speed of 10 m/s.

•  What is the final kinetic energy of the crate?

•  Where did this kinetic energy come from?

Work-Kinetic Energy Theorem •  By causing an object to speed up or slow down, you

change its kinetic energy. •  The change in kinetic energy EQUALS the amount of

work that you did!

W = ΔKE

Careful: Work is not the same thing as energy, it’s a way of transferring energy from place to

place or from one form into another form

Example: Rollercoasters

•  Where is the cart going the fastest? •  Could the cart ever get higher than

point A on its own?

Ranking: Which has the greatest kinetic energy?

Ranking: A ball is released from rest, and rolls along a nearly frictionless

track. Rank the four points in terms of their potential energy, from greatest to

least.

Ranking: A ball is released from rest, and rolls along a nearly frictionless

track. Rank the four points in terms of their kinetic energy, from greatest to

least.

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Conservation of Energy

•  Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes

Example: Free Fall revisited

•  If you drop a ball from height of 100 m, how fast is it going right before it hits the ground?

•  How much time did it spend falling?

Chapter 8: Rotational Motion

Rotational Motion

In physics we distinguish two types of motion for objects: •  Translational Motion (change of location):

Whole object moves through space. •  Rotational Motion - object turns around an

axis (axle); axis does not move. (Wheels)

Angular Position

•  We will measure angular position in revolutions:

•  Counterclockwise (CCW): positive rotation

•  Clockwise (CW): negative rotation

Linear Distance d vs. Angular distance Δθ

For a point at radius R on the wheel, d = 2πRΔθ for Δθ in revolutions

R

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Ranking: Rolling Cups

•  Which of the cups will roll in the straightest path?

•  Which of the cups will roll in the most curved path?

Angular Velocity: ω •  Avg. Angular Velocity = # Revolutions/

(Time Taken)

ω = Δθ / t •  Unit: Revolutions/s or Revolutions/min

(RPM) •  Sign convention : ω is positive for

counterclockwise rotation, negative for clockwise rotation

Tangential Velocity

•  Every spot on a rotating object has both angular velocity and tangential velocity.

•  vt = Δd/Δt = 2πRΔθ/Δt = 2πRω

R

RΔθ

Speed in Circular Motion

•  Rotational Speed ω: Rev.s per second •  Tangential speed vt: distance per second •  Two objects can have the same rotational

speed, but different tangential speeds!

Example: Gears •  Two wheels are

connected by a chain that doesn’t slip.

•  Which wheel has the higher rotational speed?

•  Which wheel has the higher tangential speed for a point on its rim?

Angular Acceleration

•  Change in angular velocity -> angular acceleration!

•  However, even if angular velocity is constant, each point also has centripetal acceleration (due to change in direction of vt)

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Simple vs. Complex Objects

Model motion with just Position

Model motion with position and Rotation

Rotational Inertia •  Rotational inertia

depends on – Total mass of the object – Distribution of the mass

relative to axis •  Farther the mass is from

the axis of rotation, the larger the rotational inertia.

•  Rotational inertia ~ (mass) x (axis_distance)2

Rotational Inertia Depends upon the axis around which it rotates •  Easier to rotate pencil

around an axis passing through it.

•  Harder to rotate it around vertical axis passing through center.

•  Hardest to rotate it around vertical axis passing through the end.

Example: Hoop vs. Disk

•  Imagine rolling a hoop and a disk of equal mass down a ramp. Which one would win?

•  Which one is “easier” to rotate (i.e., has less rotational inertia)?