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PHY131H1F Summer – Class 10 Today: •  Rotational Motion •  Rotational energy •  Centre of Mass •  Moment of Inertia •  Oscillations; Repeating

Motion •  Simple Harmonic Motion •  Connection between

Oscillations and Uniform Circular motion

•  Potential and Kinetic Energy in Oscillations

Moment of inertia is

A.  the rotational equivalent of mass. B.  the point at which all forces appear to act. C.  the time at which inertia occurs. D.  an alternative term for moment arm.

Pre-class reading quiz

Rotational Motion An object rotates about an axis. The change in angle is the same for any point on the object. Its angular velocity is

The units of ω are . If the rotation of the object is speeding up or slowing down, its angular acceleration is

The units of α are

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“Rolling Without Slipping”

When a round object rolls without slipping, the distance the axis, or centre of mass, travels is equal to the change in angular position times the radius of the object.

s =

The speed

v =

The acceleration

a =

Center of Mass

The center of mass is

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Rotation About the Center of Mass

An unconstrained object (i.e., one not on an axle or a pivot) on which there is no net force a point called the center of mass. The center of mass

while every other point in the object undergoes circular motion around it.

Rotational Energy A rotating rigid body has kinetic energy because all atoms in the object are in motion. The kinetic energy due to rotation is called rotational kinetic energy.

Here the quantity I is called the object’s moment of inertia.

The units of moment of inertia are kg m2. An object’s moment of inertia depends on the axis of rotation.

Important Concepts

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Updated Conservation of Energy…

•  A metal hoop has the same mass and radius as a wooden disk. They are both released from rest at the top of an incline, and allowed to roll down, without slipping. Which will roll faster down the incline?

A.  Metal hoop B.  Wooden disk C.  Neither; both will roll at the same speed.

Discussion Question and Demonstration

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Compare and Contrast Soup Cans •  About same

•  About same

•  Thick paste, so when this can is rolling, the contents

as one solid object, like a solid cylinder

•  Low viscosity liquid, so the can itself rolls while the liquid may just

•  Two soup cans begin at the top of an incline, are released from rest, and allowed to roll without slipping down to the bottom. Which will win?

Predict:

A.  Cream of Mushroom will win B.  Chicken Broth will win C.  Both will reach the bottom at about the same

time.

Ultimate Soup Can race: Cream of Mushroom vs. Chicken Broth!!

The Parallel-Axis Theorem

•  Suppose you know the moment of inertia of an object when it rotates about an axis which passes through the centre of mass: Icm

•  You can find the moment of inertia when it is rotating about an axis 2, (I2) which is a distance d away:

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Four Ts are made from two identical rods of equal mass and length. Rank in order, from largest to smallest, the moments of inertia Ia to Id for rotation about the dotted line.

A.  Ia > Id > Ib > Ic B.  Ic = Id > Ia = Ib C.  Ia = Ib > Ic = Id D.  Ia > Ib > Id > Ic E.  Ic > Ib > Id > Ia

(a) (b) (c) (d)

Four Ts are made from two identical rods of mass, m, and length L. Find the moments of inertia about the dotted line.

Some oscillations

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Period, frequency, angular frequency

•  The oscillation frequency f is measured in

•  We may also define an in radians per second, to describe the oscillation.

•  The time to complete one full cycle, or one oscillation, is called •  The frequency, f, is Frequency and period are related by

The Spring-Mass System

The force exerted on the mass by the spring: (Hooke’s Law) (Newton’s Second Law)

Combine to form a differential equation:

How do you solve a differential equation?? of course!

# Set the initial conditions, and the time-step:!k = 9.0!mass = 1.0!x = 0!vx = 150!t = 0!dt = 0.005!

# Run an infinite loop that shows a green ball:!While 1 == 1:! a = -(k/mass) * x! v = vx + a*dt! x = x + vx*dt! t = t + dt!

# display a green ball at the new x position:! greenBall.pos = (x, -10, 0)!

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x!

t!

Results of the Python code:!

output points from the program

A function which seems to fit

Weird coincidence noted by Knight:

This is the position graph of a mass on a spring. What can you say about the velocity and the force at the instant indicated by the dotted line?

A. Velocity is positive; force is zero. B. Velocity is negative; force is zero. C. Velocity is negative; force is to the right. D. Velocity is zero; force is to the right. E. Velocity is zero; force is to the left.

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This is the position graph of a mass on a spring. What can you say about the velocity and the force at the instant indicated by the dotted line?

A. Velocity is positive; force is zero. B. Velocity is negative; force is zero. C. Velocity is negative; force is to the right. D. Velocity is zero; force is to the right. E. Velocity is zero; force is to the left.

Simple Harmonic Motion If the initial position of an object in SHM is , then we may still use the cosine function, with

In this case, the two primary kinematic equations of SHM are:

•  An object moves with simple harmonic motion. If the amplitude and the period are both increased by a factor of 2, the object’s maximum speed is

A.  decreased by factor of ¼. B.  decreased by factor of ½. C.  increased by factor of 4. D.  increased by factor of 2. E.  unchanged.

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equilibrium

A mass hangs motionless from a spring. When the mass is pulled down and held at rest, the total energy of the mass and spring is A. larger than before. B. the same as before. C. less than before.

Gravitational Potential Energy increases with height. Elastic Potential Energy increases as a spring is stretched. Kinetic Energy increases with speed.

Discussion Question

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Before Next Class:

•  Read Chapter 14 of Knight. •  Read Chapter 15, sections 15.1 to 15.3. •  Complete MasteringPhysics.com Problem Set 8

due by June 14 at 11:59pm •  Do Suggested End-of-chapter Exercises and

Problems from Knight: Ch.12: 5, 7, 11, 17, 53 and 77