Physics 218

Alexei Safonov

Lecture 24: Simple Harmonic

Motion

Schedule

• Today – finish SHM, start review• Next lecture (Monday May 4) is the last

lecture – finish review• Final Exam:

– Friday May 8: 3.30-5.30pm

Uniform Circular Motion

It turns out Uniform Circular Motion is really simple harmonic motion but in two dimensions

Uniform Circular Motion in 2D• A ball is moving anti-clockwise in

uniform circular motion with a radius R and speed V. At t=0 we set Q0=0.

1. What is the equation of motion for Q?2. What is the equation of motion in the x

direction?3. What is the displacement at all times?4. What is the velocity at all times?5. What is the speed at all times?

Mass on a spring with g

h

Initial Final

Take an un-stretched spring. Gently add a mass and let go. The total distance from unstretched length to the new equilibrium is h. The system is then allowed to oscillate.

What is the angular frequency w of motion?

Ballistic Pendulum

• A bullet plows into a block of wood which is part of a pendulum.

• We know m, M and v.

• How high, h, does the block of wood go?

• What is the equation of motion after the collision?

A pendulum is made by hanging a thin hoola-hoop of diameter D on a small nail. What is the angular frequency of oscillation of the hoop for small displacements? (ICM = mR2 for a hoop)

The angular frequency of oscillation of the hoop for small

displacements will be given by

R X

CM

Use parallel axis theorem: I = ICM + mR2

m

= mR2 + mR2 = 2mR2

pivot (nail)

I

mgRCM

D

g

R

g

Rm

Rgm

22 2

SoD

g

Physics 218

Alexei Safonov

Final Review

Important Topics/Problem Types

• Motion with constant Acceleration, Projectile Motion: Chaps 1-3

• Uniform Circular Motion: Chaps 3, 5• Forces, Normal Forces, Friction:

Chap 4• Work, Energy and Conservation of

Energy: Chaps 6 & 7

Topics Continued

• Momentum: Chap 8• Rotational Motion: Chaps 9 & 10• Statics: Chap 11• Gravity: Chap 13• Oscillations and Vibrations: Chap 14• Math: Derivatives, Integrals,Vectors,

dot products, cross products

Kinematics

• Problem solving:– Only a few equations you can possibly write:

• x(t), y(t), vx(t), vy(t)

– There is literally nothing else, the solution must come from these equations, if it exists• Once you made sure all information given is used

in the equations, the rest is just a math exercise

d

Hunter in the Forest

• If the bullet leaves the gun at a speed of V0, by how much (vertically) will it miss the target? In other words, on the diagram what is the value of d.

• Assuming the bullet leaves at the same speed, at what angle should the gun be aimed so that the target will be hit? Hint: 2sinqcosq = sin2q

A hunter aims directly at a target which is a distance R away. Note that the gravitational acceleration near the earth’s surface is g. Ignore air friction.

Ball Thrown Off a Cliff

A ball is thrown horizontally out off a cliff of a height h above the ground. The ball hits the ground a distance D from the base of the cliff. Assuming the ball was moving horizontally at the top and ignoring air friction, what was its initial velocity?

h

Dy=0

Centripetal Acceleration

• Vector difference V2 - V1 gives the direction of acceleration a

dtvvdtvda /)(/ 12

a

RRva /2

Dynamics

• Three main types of problems;– Single object motion (trivial)– Multiple objects connected to each other

• Consider each object separately, use Newton Laws to include forces between objects

• Connect kinematics and dynamics

– Something involving circular motion• Consider kenematics and dynamics separately,

then connect the two together

2 boxes from the last timeTwo boxes with masses m1 and m2 are placed on a

frictionless horizontal surface and pulled with a Force FP. Assume the string between doesn’t stretch and is massless.

a)What is the acceleration of the boxes?

b)What is the tension of the strings between the boxes?

M2 M1

An Incline, a Pulley and two BoxesIn the diagram

given, m1 and m2 remain at rest and the angle Q is known. The coefficient of static friction is m and m1

is known. What is the mass m2?Is it a single value or a range?

Q

m2m

1

Ignore the mass of the pulley and cord and any friction associated with the pulley

Banking AngleYou are a driver on the

NASCAR circuit. Your car has m and is traveling with a speed V around a curve with Radius R

What angle, Q, should the road be banked so that no friction is required?

If there is friction (m is given), what’s max speed you can drive with?

Conical PendulumA small ball of mass m

is suspended by a cord of length L and revolves in a circle with a radius given by r = LsinQ.

1. What is the velocity of the ball?

2. Tension FT?3. Calculate the period

of the ball.

Work/Energy

• Figure out what forces do work, which don’t

• Get potential forces separate and use potential energy instead of work– Gravity, springs

• Do before and after– Find whatever has been asked in the problem

Friction and SpringsA block of mass m is traveling on a rough surface. It reaches a spring (spring constant k) with speed vo and compresses it by an amount D. Determine m

Loop-the loop• You are building a loop-the-loop want to know

the minimum height of point A (in the figure below) such that a car slides along the track without falling off at the top (point B). The car starts from rest at point A at eight H above the bottom of the loop and slides without friction. The car has mass m; air resistance is negligible; the loop has radius known R.• Draw a free-body diagram for the car at

point B when the car moves around the loop safely.

• Express the minimum value of height H (in terms of R) such that the car moves around the loop without falling off at the top (point B)?

• If H=4R, what is the normal force of the track at the top of the loop?

H

Ski Vacation• You take a ski vacation. While at the summit of a large

slope, you start from rest and then ski over two successively lower hills. The lowest hill is essentially a semi-circle of height H centered at the 0 level. You want to leave the lowest hill at its top and fly through the air. Assume no friction and air resistance. Magnitude of gravity = g.

H

h

Start here

Start flying– How far up the slope

must you be when you start down the hill? Put your answer in terms of g and H2.

Pendulum

• You fire a bullet of mass M1 and unknown speed directly into a wooden block of mass M2. The block is suspended by wires from the ceiling and is initially at rest. The bullet embeds in the block and the block swings up to a maximum height (Hmax) above its initial position. Magnitude of gravity = g. Ignore air friction.

• What is the speed of the bullet, when fired? • What is the change in mechanical energy after the bullet

gets stuck in the block?• Is this an elastic or inelastic collision? Why?

Mass Moving in a Circle• A puck of mass M, attached to a massless string, moves

in a circle of constant radius R about a small hole on a frictionless, horizontal table. The end of the string is connected to a small block of mass m through the hole as shown. The block and puck may be treated as particles.– Draw the force diagram for the puck including the magnitude

and direction of each force.– Find the speed (v1) of the puck in terms of g, R, M and m– A woman pulls the block (of mass m) down by ½R. Find the

resultant speed, v2, of the rotation in terms of v1. Hint: Use conservation of angular momentum

Rotational Dynamics

• Conservation laws help in many problems– Energy

• Remember to include rotational kinetic energy!

– Momentum• Otherwise, two equations:

– Sum of all Forces = ma– Sum of all torques = Ia

• Center of mass is always a safe choice for the origin

Ice Skating• You decide to take up ice-skating and want to learn all the

moves. You are proud of the fact that you have learned to spin in place on the ice. Your friends video tape you spinning and then bringing your arms in close to your body. From the tape you are able to measure your initial and final moments of inertia, I0, and If = Io/3, as well as your final angular speed, wf. In terms of I0 and wf, find:– Your final angular momentum– Your initial angular speed– Your initial kinetic energy – How much work you have to do to bring in your arms.

• At the end of your spin, you put your foot down and dig your toe into the ice and come to a stop. You measure the time it takes to stop to be t seconds. Assuming constant negative angular acceleration: – What is the torque your toe exerts?

Yo-Yo

• A primitive yo-yo is made by wrapping a string several times around a thin-walled hollow cylinder with mass M = 100 g. You hold the end of the string stationary while releasing the cylinder with no initial motion. The string unwinds but does not slip or stretch as the cylinder drops and rotates. In the absence of air resistance, find– the tension force in the string;– the linear acceleration of the cylinder;– the speed of the center of mass of the cylinder after it

has dropped a distance h = 50 cm.

Ball Rolling Down a Hill

• You are the technical advisor on a new movie. The script has a giant spherical ice ball, I = 2/5 mr2, where you have measured m and r. It rolls down a hill (without slipping) starting from rest right at a person who is at the bottom. You know the length of the slope (L) and its angle to the horizontal (q) and the ball starts from the top of the hill

• Your job is to calculate the speed of the ice ball at the bottom of the hill (you want to make sure it’s fast enough to look cool) and

• The time it takes to get down the hill so the “victim” knows exactly when to dive out of the way.

Statics

• If you know dynamics of the rotational motion, statics is very easy

• Write equations for forces and torques and set right hand side to zero– There are some tricks, e.g. picking a good

origin, but they just make the solution easier. Any origin should give you the same answer.

Ladder

• A uniform ladder 6 m long rests against a frictionless, vertical wall at an angle 450 above the horizontal. The ladder weighs 200 N. A man with mass 100 kg climbs slowly up the ladder. The ladder starts to slip when the man is at a height h = 2 m above the ground. For this moment of time, find – the normal force that the ground exerts on the ladder;– the normal force that the wall exerts on the ladder;– the coefficient of static friction between the foot of the

ladder and the ground.

Gravity

• To first order, all these problems are a mix of circular motion and dynamics– Chapters 3 and 5

• The only two extra things:– Need to know how to calculate the

gravitational force– Need to know Keppler’s laws in case the

problem deals with a non-circular orbit.

Apollo 11

• The Apollo 11 spacecraft, mass m, orbited the moon in uniform circular motion at a radius R. The period of the orbit was T.– Find the orbital speed of the spacecraft– Find the magnitude and direction of the

acceleration of the spacecraft.– Find the mass of the moon in terms of the

variables given

Simple Harmonic Motion

A block of mass m sits on a frictionless surface attached to a spring with spring constant k. We can write down

• Hooke’s law: F=-kx• Newton’s law: F = ma =md2x/dt2.

• Given that the general equation of motion for the position, x, as a function of time t is x(t) = Asin(wt+f), show that this equation is a solution of Hooke’s law and Newton’s law with w2= k/m.

• At time t=0, the spring is compressed to the position -x0 and released from rest. Using your previous results, find the amplitude. You may NOT just write down the solution, rather you must show it to be true mathematically.

• For the same system, find the phase of the motion

That’s it…

• It’s been a pleasure• I hope you’ve enjoyed it as well and

perhaps even learned something ;)• Best of luck on the final…