ENERGY, POWER and MOMENTUM

Dr. Johnny B. Holmes

## Introduction

In this third part of the course we consider the concepts of work, energy, power, momentum, and angular momentum.

First we introduce the concepts of work and energy. We look at two particular kinds of energy: kinetic (or motional) energy and potential (positional) energy. We also look at one of the most powerful tools in physics: the law of Conservation of Energy. Finally, we consider the concept of power.

Next we consider collisions and introduce the concept of momentum to help in our analysis. This leads to the formulation of another important law: Conservation of Momentum.

We then again consider the concept of torque but this time consider its relation to energy. We also introduce the concept of angular momentum and the law: Conservation of Angular Momentum.

## Work and Energy

1. Work Z
1. definition: exert a force through a distance: Work = F · s
2. units: Joule = Newton - meter
3. dot product: work is a scalar with magnitude: W = F s cos(q Fs)
4. most general formula involves calculus integration: W = ò F cos(q ) ds (must put in limits!)
2. Energy
1. definition: the capacity to do work (under ideal conditions)
2. units: joules
3. kinetic energy: energy due to motion: KE = 1/2mv²
4. potential energy: energy due to position
1. gravitational (near earth): PE = mgh
2. gravitational (in general): PE = -Gm1m2/r
3. spring potential energy: PE = 1/2kx2
3. Conservation of energy AA,BB,CC
1. without frictional losses
2. with losses
3. with work added or subtracted
4. Power DD,EE
1. definition: Power = dW/dt ; since W = F·s then P = F·v
2. units: Watt = Joule/sec
3. cost of utilities (pay for energy, not force or power)

Z. How much work is done in lifting a car of mass 1500 kg up six feet (so the mechanic can work underneath it)?

AA. What is the escape velocity (a) for the earth (Mass of earth is 6 x 1024kg, radius is 6371 km.)? (b) for the moon (Mass of moon is 7.34 x 1022kg, radius is 1741 km.)?

BB. Consider problem U concerning the geosynchronous satellite of mass 50 kg. It orbits at a radius of 42,300 km with a speed of 3076 m/s. a) How much energy is needed to lift the satellite from the earth's surface (Re=6,370 km) up to that radius? b) How much kinetic energy does the satellite have as it orbits?

CC. A sled of mass 5 kg is at the top of a hill 20 meters in vertical height above the base of the hill. The hill has a constant grade of 30° with the horizontal. (a) If there is no friction between the sled and the ground, how fast will the sled be going when it reaches the base if it has started from rest? (b) If there is a coefficient of friction between the sled and the ground of 0.1, how fast will the sled be going when it reaches the base if it has started with a speed of 5 m/sec? (c) If the sled has a mass of 10 kg, would the answers to (a) and (b) change?

DD. A car of mass 1700 kg accelerates from rest to a speed of 25 m/s in a time of 13 seconds.
a) What is the final kinetic energy of the car?
b) What is the average power generated by the car during the 13 seconds?
c) If the power is constant during the 13 seconds, does the acceleration remain constant also, decrease or increase as the speed increases?

EE. The average power output of the sun per unit area above the atmosphere is 1.35 kW/m². However, the average power per area reaching the ground (including day-night, clear-cloudy, summer-winter) is about 230 Watts/m². If 8 square meters of solar collectors are mounted on a house and the collectors are 15% efficient, and if the cost of energy is \$.06/(Kw*hr), how much money will the collectors save in one month? (There are several other factors that need to be considered to make this analysis more applicable such as storage of the energy for use when its needed and capital costs of the collector.)

Z. 26,883 Joules.

AA. (a) 11200 m/s = 25000 mph; (b) 2370 m/s = 5300 mph.

BB. 2.67 x 109 Joules; 2.37 x 108 Joules.

CC. (a) 19.8 m/s = 45 mph; (b) 18.7 m/s = 42 mph; (c) No.

DD. (a) 531,000 Joules; (b) 40,900 Watts = 54.8 hp; (c) decreases.

EE. \$11.92 .

## Momentum: useful for collisions and explosions

1. Force and momentum   FF,GG,HH
1. Newton's second law: SF = dp/dt
2. Impulses: Dpx = Fx-avg Dt
2. Conservation of Momentum (only if neglect external forces!): Spx-init = Spx-final
3. Collisions          II
1. elastic (e.g., rubber bullet): Elost = 0
1. 1-D: two eqs. (cons of E and p), can have two unknowns
2. 2-D: three eqs. (cons of E, px and py), can have three unknowns
2. inelastic: Elost > 0 .
1. becomes embedded (e.g., lead bullet) v1-final = v2-final
2. goes through (e.g., steel bullet) no special information
4. Explosions: collisions in reverse             JJ
1. guns: v1-init = v2-init; Elost > 0 !
2. rockets: treat as controlled, continuous explosion

FF. A constant net force of 5 Nt in the x-direction is applied to an object of mass 10 kg for 6 sec. What is (1) the total change of momentum, (2) the final velocity, and (3) the energy gain or loss of the object...(a) if the object started at rest? (b) if the object was going at a speed of 4 m/sec in the positive x-direction? (c) if the object was going at a speed of 4 m/sec in the negative x-direction? (d) if the object was going at a speed of 4 m/sec in the positive y-direction?

GG. What causes the differences in the answers to the different parts of problem FF above?

HH. An unlucky bystander finds himself in the center of a shootout between the good guys and the bad guys. A 5.0 gm bullet moving at 100 m/sec strikes him and lodges in his shoulder. Assuming the bullet undergoes uniform deceleration and stops in 6.0 cm, find (a) the time taken to stop, (b) the impulse on the shoulder, and (c) the average force experienced by the man.

II. Consider a 5 gm bullet at a speed of 150 m/sec in the positive x-direction that then hits a 2 kg target which was initially at rest. What is the speed of the target after it is hit by the bullet (a) if the bullet is steel and becomes imbedded in the target? (b) if the bullet is rubber and bounces off at 180° from the initial path (assume elastic collision) ? (c) what is the final speed of the bullet in part-a and part-b ?

JJ. An astronaut during a spaceship flight had her safety line cut by a sharp edge and finds herself floating beside the ship. (a) Can she "swim" back to the spaceship? (b) If she has a wrench of 2 kg in her hand and throws this away from the ship at 20 m/sec, can she reach the ship? (c) If she is 25 meters from the spaceship, how long will it take her to reach the ship after she throws the wrench assuming the astronaut (plus suit, etc.) has a mass of 70 kg? (d) What is the astronaut's weight in pounds?

FF.       a-1] (30 kg*m/s,0);      a-2] (3 m/s,0); a-3] 45 J;

b-1] (30 kg*m/s,0);      b-2] (7 m/s,0); b-3] 165 J;

c-1] (30 kg*m/s,0);      c-2] (-1 m/s,0);            c-3] -75 J;

d-1] (30 kg*m/s,0);      d-2] (3 m/s, 4 m/s); d-3] 45 J.

GG. The same force and time imply the same change in momentum, but the different initial velocities cause a different distance through which the force acts causing different energy changes.

HH. (a) 1.2 millisec; (b) 0.5 kg*m/s; (c) 416.67 Nt.

II. (a) 0.374 m/s; (b) 0.748 m/s; (c) 0.374 m/s, 149.25 m/s

JJ. (a) no; (b) yes; (c) v = 0.57 m/s, t = 43.75 s; (d) in outer- space his weight would be zero, on the Earth it would be 154 pounds.

## Rotational Dynamics

1. Review of uniform circular motion         KK,LL
1. angles - definition & units (q radians = s/r where s is arclength)
2. circular motion at constant speed: vT = w r, aR = w ²r
3. Newton's 2nd law and circular motion: S FR = maR
4. satellites (FR = G MEarthmsat/r²)
2. Circular motion             MM,NN
1. angle as vector s = q radiansr (s is arclength) (from q = s/r)
2. angular velocity: wavg = Dq /Dt (rad/sec); vT = wr; vR = 0
3. angular acceleration vector: a avg = Dw /Dt (rad/sec²)
4. tangential and radial components: aT = a r; aR = w ²r
3. Special case: uniform a : q = qo +wot + 1/2a t²; w = wo + a t
4. Torque and rotation
1. torque - rotational analogue of force
2. torque as vector: t = r F sinq (q = angle between r and F)
3. moment of inertia - rotational inertial: I = S mr²
5. Rotational energy          OO
1. kinetic energy for rotation: KE = 1/2Iw ² (similar to KE=1/2mv²)
2. work from torque: W = t q
3. power from torque: P = dW/dt = tw
4. total KE: translation plus rotation: KE = 1/2mv² + 1/2Iw²
6. Angular momentum: L = r x p
1. magnitude: L = r m v sin(q rv) = I w (similar to p=mv)
2. direction: right hand rule: curl fingers in w , thumb points in L
7. Changing angular momentum: S t = dL/dt
1. if S t external = 0, then have Conservation of L: S Linit = S Lfinal
2. if S t is parallel to L, then change magnitude of L
3. if S t is perpendicular to L, then change direction of L (get precession)

KK. The diameter of Jupiter is 144,000 km and at its nearest, it is 6.26 X 108 km from the earth. What is the angle that Jupiter makes at the eye (a) in radians? (b) in degrees? (c) in seconds of angle?

LL. (a) How fast are you going right now due to the fact that the earth is rotating about its axis? (Memphis is at about 35° N lattitude, and the radius of the earth is 6,400 km.) (b) How fast are you going right now due to the fact that the earth is orbiting the sun? (The radius of the earth's orbit is 1.49 * 108 km.)

MM. A man is at the earth's equator. In terms of N, E, S, W, up, and down, what is the direction of the following due to the earth's rotation about its axis: a) w ? b) a ? c) v? d) a?

NN. A car accelerates uniformly from rest to a speed of 15 m/s in a time of 20 seconds on wheels of radius 30 cm. Find the angular acceleration of one of its wheels (a) at t = 0 sec, and (b) at t = 20 sec; (c) Find the number of revolutions turned by the wheel in the process. (d) What is the angular speed of the wheel at t = 0 sec; and (e) at t = 20 sec? (f) What is the tangential acceleration (due to spinning only) of a point on the outside part of the wheel at t = 0 sec; (g) at t = 20 sec? (h) What is the radial acceleration of a point on the outside part of the wheel at t = 0 sec? (i) at t = 20 sec?

OO. A spherical ball of mass 250 grams and radius 4 cm rolls (without slipping) down an incline that has one end raised 1 meter above the ground and the other end is on the ground. The incline itself is 3 meters long. (a) What is the initial potential energy of the ball if it is at the top of the incline? (b) As the ball rolls down (without slipping), is there any energy lost to friction? (c) Ignoring air resistance, how fast will the ball be going after it rolls all the way down to the ground? (d) How fast will the ball be rotating? (e) If the ball slid with minimal friction (e.g., icy surface), will the ball be going faster or slower than the case of part c above?

KK. (a) 2.3 x 10-4 rad; (b) .013°; (c) 47.4 seconds.

LL. (a) 1370 km/hr or 850 mph; (b) 106,800 km/hr or 66,750 mph.

MM. (a) N; (b) none since a is zero; (c) E; (d) down.