PC1431 MasteringPhysics Assignment 6

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Assignment 6: Dynamics of Rotational Motion

Due: 2:00am on Saturday, October 23, 2010

Note: To understand how points are awarded, read your instructor's Grading Policy. [Switch to Standard Assignment View]

Finding Torque

A force of magnitude , making an angle with the x axis, is applied to a particle located at point A,

at Cartesian coordinates (0, 0) in the figure. The vector and the four reference points (i.e., A, B, C, and D) all lie in the xy plane. Rotation axes A - D lie parallel to the z axis and pass through each

respective reference point.

The torque of a force acting on a particle having a position vector with respect to a reference point (thus points from the reference point to the point at which the force acts) is equal to the cross product of and , . The magnitude of the torque is , where is the angle between and ; the direction of is perpendicular to both and . For this problem ; negative torque about a reference point corresponds to clockwise rotation. You must express in terms of , , and/or when entering your answers.

Part A

What is the torque due to force about the point A? Hint A.1 When force is applied at the pivot point

Hint not displayed

Express the torque about point A at Cartesian coordinates (0, 0). ANSWER:

= 0

Correct

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What is the torque due to force about the point B? (B is the point at Cartesian coordinates (0, ), located a distance from the origin along the y axis.)

Hint B.1 Finding with respect to a reference point

Express the torque about point B in terms of , , , , and/or other given coordinate data. ANSWER:

=

Correct

Part C

What is the torque about the point C, located at a position given by Cartesian coordinates ( , 0), a distance along the x axis?

Hint C.1 Clockwise or counterclockwise?

Express the torque about point C in terms of , , , , and/or other given coordinate data. ANSWER:

=

Correct

Part D

What is the torque about the point D, located at a distance from the origin and making an angle with the x axis?

Express the torque about point D in terms of , , , , and/or other given coordinate data. ANSWER:

=

Correct

Note that the cross product can also be expressed as a third-order determinant

which simplifies to when and lie in the xy plane.

An Unfair Race

This applet shows the results of releasing a frictionless block and a rolling disk with equal masses from the top of identical inclined planes.

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the top of identical inclined planes. Part A

Which of the following is the best explanation of the results shown in the applet? ANSWER:

The disk loses energy to friction as it rolls, but the box is frictionless and so it speeds up more quickly and gets to the bottom first.

The potential energy of the disk is converted into translational and rotational kinetic energy, so the translational speed grows more slowly than that of the box, which has no rotational energy.

The net forces on the two objects are equal, but the force on the disk gets partially used up in creating the torque necessary to make it roll. The net forces on the two objects are equal, but the force on the disk is not directed parallel to the ramp, and so does not create as great an acceleration down the ramp.

Correct

This applet shows the same situation, but it also shows, through bar graphs that change with time, the way that the energy is transformed as the box and the disk go down the inclined plane.

Assume that the box and disk each have mass , the top of the incline is at height , and the angle between the incline and the ground is (i.e., the incline is at an angle above the horizontal). Also, let the radius of the disk be .

Part B

How much sooner does the box reach the bottom of the incline than the disk? Hint B.1 How to approach the problem

Hint not displayed Hint B.2 Find the final speed of the box

Hint not displayed Hint B.3 Find the final speed of the disk

Hint not displayed Hint B.4 Finding the average speed

Hint not displayed Hint B.5 Finding the time from the average speed

Hint not displayed Hint B.6 Find the length of the incline

Express your answer in terms of some or all of the variables , , , and , as well as the acceleration due to gravity .

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acceleration due to gravity . ANSWER:

Correct

You should look at your answer and consider limiting cases. A simple one is that the time difference should tend to zero as the length of the board shrinks to zero. Simply express the height of the board in terms of the length of the incline and you'll see that your answer indeed behaves this way.

Your answer also predicts that the difference in time grows longer as shrinks toward zero while the height remains fixed (i.e., the difference in time grows longer as the length of the board grows longer). It might not be immediately obvious to you that this should happen, but it is not inconceivable, and you can do some simple experiments to see that it is actually true.

As grows toward , you might expect the difference in time to go to zero, because if you drop a disk and a box they fall at the same rate. However, recall that your derivation included the assumption that the disk rolls without slipping, which is definitely not the case if the disk is simply dropped vertically. Therefore, this formula shouldn't apply to the case of simply dropping the disk and box. Can you think of a situation with a vertical drop in which the disk would obey ?

Pulling a String to Accelerate a Wheel

A bicycle wheel is mounted on a fixed, frictionless axle, as shown . A massless string is wound around the wheel's rim, and a constant horizontal force

of magnitude starts pulling the string from the top of the wheel starting at time when the wheel is not rotating. Suppose that at some later time the string has been pulled through a distance . The wheel has moment of inertia

, where is a dimensionless number less than 1, is the wheel's mass, and is its radius. Assume that the string does not slip on

the wheel.

Part A

Find , the angular acceleration of the wheel, which results from pulling the string to the left. Use the standard convention that counterclockwise angular accelerations are positive.

Hint A.1 Relate torque about the axle to force applied to the wheel

Hint not displayed Hint A.2 Relate torque on wheel to angular acceleration

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Express the angular acceleration, , in terms of , , , and (but not ). ANSWER:

=

Correct

Part B

The force pulling the string is constant; therefore the magnitude of the angular acceleration of the wheel is constant for this configuration.

Find the magnitude of the angular velocity of the wheel when the string has been pulled a distance .

Note that there are two ways to find an expression for ; these expressions look very different but are equivalent.

Hint B.1 What the no-slip case means

Hint B.2 Review of translational motion with constant acceleration

In kinematics, you learned that for translational motion with constant acceleration the velocity is given by . The wheel is stationary at , so the displacement of the string, , will be proportional to .

Hint B.3 When has the string been pulled a distance ? Find the time when the string has been pulled a distance . Express your answer in terms of and .

ANSWER:

=

Correct

Hint B.4 Relating translational acceleration and angular acceleration Find the magnitude, , of the acceleration of the string.

Express your answer in terms of and . ANSWER:

=

Correct

There is no slip, so the magnitude of the velocity of the string is given by . Take the first time derivative of this equation to relate and .

Express the angular velocity of the wheel in terms of the displacement , the magnitude of the applied force, and the moment of inertia of the wheel , if you've found such a

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of the applied force, and the moment of inertia of the wheel , if you've found such a solution. Otherwise, following the hints for this part should lead you to express the angular velocity of the wheel in terms of the displacement , the wheel's radius , and .

ANSWER:

=

Correct

This solution can be obtained from the equations of rotational motion and the equations of motion with constant acceleration. An alternate approach is to calculate the work done over the displacement by the force and equate this work to the increase in rotational kinetic energy of rotation of the wheel

Part C

Find , the speed of the string after it has been pulled by over a distance . Hint C.1 Relating the speed of the string to the angular velocity

Express the speed of the string in terms of , , , and ; do not include , , or in your answer.

ANSWER:

=

Correct

Note that this is the speed that an object of mass (which is less than ) would attain if pulled a distance by a force with constant magnitude .

Hoop on a Ramp

A circular hoop of mass , radius , and infinitesimal thickness rolls without slipping down a ramp inclined at an angle with the horizontal.

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Part A

What is the acceleration of the center of the hoop? Hint A.1 How to approach the problem

Hint not displayed Hint A.2 Find the torque about the center of mass

Hint not displayed Hint A.3 Find an expression for the torque

Hint not displayed Hint A.4 What is the moment of inertia of the hoop?

Hint not displayed Hint A.5 Find the frictional force

Hint not displayed Hint A.6 Find the linear acceleration

Hint not displayed Hint A.7 Putting it all together

Express the acceleration in terms of physical constants and all or some of the quantities , , and .

ANSWER:

=

Correct

So the acceleration is independent of the hoop characteristics, that is, the mass and size (radius) of the hoop. This is quite generally true for objects freely rolling down a ramp; the acceleration depends only on the distribution of mass, for example, whether the object is a disk or a sphere, but within each class the acceleration is the same. For example, all spheres will accelerate at the same rate, though this rate is different from the rate for (all) disks.

Part B

What is the minimum coefficient of (static) friction needed for the hoop to roll without slipping? Note that it is static and not kinetic friction that is relevant here, since the bottom point on the wheel is not moving relative to the ground (this is the meaning of no slipping).

Hint B.1 How to approach the problem

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Hint B.2 Find the maximum value of the frictional force

Hint not displayed Hint B.3 What is the normal force?

Hint not displayed Hint B.4 Putting it all together

Express the minimum coefficient of friction in terms of all or some of the given quantities , , and .

ANSWER:

=

Correct

Part C

Imagine that the above hoop is a tire. The coefficient of static friction between rubber and concrete is typically at least 0.9. What is the maximum angle you could ride down without worrying about skidding?

Express your answer numerically, in degrees, to two significant figures. ANSWER:

= 61

Correct

When roads are wet or icy though, the coefficient of friction between rubber and concrete drops to about 0.3 (or less), making skidding likely at much smaller angles.

Acceleration of a Pulley

A string is wrapped around a uniform solid cylinder of radius , as shown in the figure . The cylinder can rotate freely about its axis. The loose end of the

string is attached to a block. The block and cylinder each have mass . Note that the positive y direction is downward and

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Part A

Find the magnitude of the angular acceleration of the cylinder as the block descends. Hint A.1 How to approach the problem

Hint not displayed Hint A.2 Find the net force on the block

Hint not displayed Hint A.3 Find the net torque on the pulley

Hint not displayed Hint A.4 Relate linear and angular acceleration

Hint not displayed Hint A.5 Putting it together

Express your answer in terms of the cylinder's radius and the magnitude of the acceleration due to gravity .

ANSWER: ₌ _{Answer not displayed}

Change in Angular Velocity Ranking Task

A merry-go-round of radius , shown in the figure, is rotating at constant angular speed. The friction in its bearings is so small that it can be ignored. A sandbag of mass is dropped onto the merry-go-round, at a position designated by . The sandbag does not slip or roll upon contact with the merry-go-round.

Part A

Rank the following different combinations of and on the basis of the angular speed of the merry-go-round after the sandbag "sticks" to the merry-go-round.

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after the sandbag "sticks" to the merry-go-round.

Hint A.1 How to approach the problem

Hint not displayed Hint A.2 Determining the change in moment of inertia

Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER:

Answer not displayed

Balancing Torques Ranking Task

A sign is to be hung from the end of a thin pole, and the pole supported by a single cable. Your design firm brainstorms the six scenarios shown below. In scenarios A, B, and D, the cable is attached halfway between the midpoint and end of the pole. In C, the cable is attached to the mid-point of the pole. In E and F, the cable is attached to the end of the pole.

Part A

Rank the design scenarios (A through F) on the basis of the tension in the supporting cable. Hint A.1 How to approach the problem

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Hint A.2 The mathematical relationship

Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER:

Answer not displayed

Twirling a Baton

A majorette in a parade is performing some acrobatic twirlings of her baton. Assume that the baton is a uniform rod of mass 0.120 and length 80.0 .

Part A

Initially, the baton is spinning about a line through its center at angular velocity 3.00 . What is its angular momentum?

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Hint A.1 Angular momentum for a rigid body rotating about an axis of symmetry

Hint not displayed Hint A.2 Moment of inertia

Express your answer in kilogram meters squared per second. ANSWER:

Answer not displayed

Part B

Part not displayed

Hockey Stick and Puck

A hockey stick of mass and length is at

rest on the ice (which is assumed to be

frictionless). A puck with mass hits the stick a distance from the middle of the stick. Before the collision, the puck was moving with speed in a direction perpendicular to the stick, as indicated in the figure. The collision is completely inelastic, and the puck remains attached to the stick after the collision.

Part A

Find the speed of the center of mass of the stick+puck combination after the collision. Hint A.1 Which conservation law to use

Hint not displayed Hint A.2 Calculate the initial momentum of the system

Hint not displayed Hint A.3 Calculate the final momentum of the system

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ANSWER: ₌ _{Answer not displayed}

Part B

After the collision, the stick and puck will rotate about their combined center of mass. How far is this center of mass from the point at which the puck struck? In the figure, this distance is .

Hint B.1 Distance from middle of stick to center of mass of stick+puck

ANSWER: ₌

Part C

What is the angular momentum of the system before the collision, with respect to the center of mass of the final system?

Hint C.1 Formula for angular momentum

Express in terms of the given variables.

ANSWER: ₌

Part D

What is the angular velocity of the stick+puck combination after the collision? Assume that the stick is uniform and has a moment of inertia about its center.

Hint D.1 How to approach the problem

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Hint D.2 Express angular momentum in terms of moment of inertia and velocity

Hint not displayed Hint D.3 Calculate the moment of inertia

Hint not displayed Hint D.4 Putting it all together

Your answer for should not contain the variable . ANSWER: ₌ _{Answer not displayed}

Part E

Which of the following statements are TRUE? 1) Kinetic energy is conserved.

2) Linear momentum is conserved.

3) Angular momentum of the stick+puck is conserved about the center of mass of the combined system.

4) Angular momentum of the stick+puck is conserved about the (stationary) point where the collision occurs.

Hint E.1 About conservation of angular momentum

Hint not displayed ANSWER: 1 only 2 only 3 only 4 only 1 & 2 1 & 4 2 & 4 1 2 & 3 2 3 & 4

The last question is optional. If your lecturer did not go through this topic, you can skip it totally.

A Toy Gyroscope

The rotor (flywheel) of a toy gyroscope has mass 0.140 kilograms. Its moment of inertia about its axis is kilogram meters squared. The mass of the frame is 0.0250 kilograms. The gyroscope is supported on a single pivot with its center of

mass a horizontal distance 4.00 centimeters from the pivot. The gyroscope is precessing in a

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from the pivot. The gyroscope is precessing in a horizontal plane at the rate of one revolution in 2.20 seconds.

Part A

Find the upward force exerted by the pivot. Hint A.1 Precession in a gyroscope

Hint not displayed Hint A.2 How to approach the problem

Hint not displayed Hint A.3 Balance of forces

Enter your answer in newtons to four significant figures. ANSWER: ₌

Part B

Find the angular speed at which the rotor is spinning about its axis, expressed in revolutions per minute.

Hint B.1 How to approach the problem

Hint not displayed Hint B.2 How to calculate the angular momentum

Hint not displayed Hint B.3 Calculate the precession angular speed

Hint not displayed Hint B.4 Calculate the torque

Hint not displayed Hint B.5 Using the angular momentum

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Enter your answer in revolutions per minute to four significant figures. ANSWER: ₌

Score Summary:

Your score on this assignment is 99.8%.