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PHY131H1S – Class 26Last class! Today:

• Review of Torque • Review of Simple Harmonic Motion• Review of the Gravitational field

Marks are being posted on portal as they come in.

Final exam is May 1 at 2pm in EX300

Section 12.5: Torque

• Torque is the rotational equivalent of force.

• A net torque on an object will give it angular acceleration, α [rad/s2]

• For an object in equilibrium, the torques must balance (net torque = 0)

A. Torque on the handle is greaterB. Torque on the metal is greaterC. Torques are the same

A common bolt-cutter has a long handle where you apply force far from the pivot, and short cutting blades, which cut the bolt or lock close to the pivot. When cutting a piece of metal, compare the torque applied to the handle to the torque applied to the piece of metal.

A. Force on the handle is greaterB. Force on the metal is greaterC. Forces are the same

A common bolt-cutter has a long handle where you apply force far from the pivot, and short cutting blades, which cut the bolt or lock close to the pivot. When cutting a piece of metal, compare the force applied to the handle to the force applied to the piece of metal.

Gravitational torque

• Whenever you calculate torque, you need to know where on an object the force is applied, relative to the pivot.

• But gravity acts on every part of the object equally!• We can calculate gravitational torque as if all its mass were

concentrated at the centre of mass

A non-equilibrium problem involving torque

• Pr. 12.70b. Find the acceleration of m1. Assume the pulley is a uniform disk with mass mp and radius R.

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Simple Harmonic Motion: Restoring Force provided by Hooke’s Law

x,v,a for Simple Harmonic Motion

x

tP

A mass attached to a spring oscillates back and forth as indicated in the position vs. time plot above. At point, P, the mass has

A. positive velocity and positive acceleration.B. positive velocity and negative acceleration.C. negative velocity and positive acceleration.D. negative velocity and negative acceleration.E. positive velocity and zero acceleration.

Simple Harmonic Motion notes…• S.H.M. is not constant acceleration, or constant force – both

vary with time.• S.H.M. results when restoring force is proportional to

displacement. Other types of oscillatory motion are possible, but not discussed in this course.

• Angular frequency ω = 2π/T, where T = period. (T = 2π/ω)

• “frequency” f = 1/T (in Hertz)

Post-class questionnaire

• Worth an entire class of clicker questions.

• Please answer TWO of the four questions on the back of the page. If your last name as shown on your student card begins with the letter – A through L, please answer questions 3 and 4

only.

– M through Z, please answer questions 1 and 2 only.

Pre/post class questionnaire, question 1.

• A car, shown below is traveling from left to right in the diagram at a constant speed. It is turning on a curved path, as shown by the dashed line. Draw a vector labelled v for velocity of the car. Draw a vector labelled F for the force of the road upon the car.

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Pre/post class questionnaire, question 2.• A carnival has a game which supposedly measures

strength. When you pound on one end of a lever with a giant hammer, the lever tosses a metal ball up a tall, vertical track toward a bell located 15 feet above you. As the ball rises upward on the frictionless track, it experiences

A. a constant upward force that depends only on the inertia of the ball as it rises toward the bell.

B. an upward force that decreases steadily, but yet remains substantially greater than zero even when the ball hits the bell.

C. an upward force that decreases steadily and reaches zero exactly at the moment the ball hits the bell.

D. no upward forces.

Pre/post class questionnaire, question 3.A fish floats motionless below the surface of a lake.

What is the direction and amount of force the water exerts on it?

A. ZeroB. Down, equal to the weight of the fishC. Up, equal to the weight of the fishD. Not enough information is given to determine.

Pre/post class questionnaire, question 4.The darker part of the hammer is made of metal, and the

white part of the hammer is made of wood. Compare the mass of the part of the hammer to the left of the dashed line (mass on the left) to the mass of the part of the hammer to the right of the dashed line (mass on the right).

A. The mass on the left is equal to the mass on the right.B. The mass on the left is greater than the mass on the

right. C. The mass on the left is less than the mass on the right.

The Gravitational Field

• Recall from section 13.5, eq.13.15. When a mass m2 is near a mass m1, it has a potential energy, given by

rmGmU g

21−=

• If m2 is much smaller than m1, we can think of m2 as a “test particle”.

• No matter where we place m2, there is a gravitational potential energy at every point in space due to m1.

• We can think of this as a property of the space itself: the gravitational potential field.

• This is a scalar field: a number is associated with every (x,y,z) point in space.

The Gravitational Field• Recall from section 11.6, eq.11.28: The Force on an

object is the negative of the gradient of its potential energy.

rr

mGmrr

UUF g

gg ˆˆ 221−=

∂∂

−=∇−=rr

• If m2 is much smaller than m1, we can think of m2 as a “test particle”.

• No matter where we place m2, there is a gravitational force at every point in space due to m1, which is the negative gradient of the potential energy.

• We can think of this as a property of the space itself: the gravitational force field.

• This is a vector field. A vector is associated with every (x,y,z) point in space.

The Gravitational Field• Equipotential surfaces map where the potential energy is

constant.

• An equipotential around the Earth is not quite spherical. It has bumps (red regions) and dips (blue regions)

• The gravitational force vectors are perpendicular to the equipotential surfaces.

From Knight Ch.13, pg.393

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Final Exam• Friday May 1, 2-4pm, EX300• Aids allowed:

– A non-programmable calculator without text storage. – A single, hand-written aid-sheet prepared by the

student, no larger than 8.5”x11”, written on both sides.

• There are 12 multiple-choice questions, worth 2 points each, or altogether 24 points.

• There are a set of four free-form questions worth 16 points, for which fully worked solutions are required.

• The total possible number of points is 40.• Good luck! Feel free to contact me by email,

phone or in person between now and May 1, and beyond!