Unwinding Cylinder - Physics and Astronomy at TAMU · Unwinding Cylinder Description: Using ... A cylinder with moment of inertia about its center of mass, mass , and radius has a

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Chapter 10Due: 11:59pm on Sunday, November 6, 2016

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Unwinding CylinderDescription: Using conservation of energy, find the final velocity of a "yoyo" as it unwinds under the influence of gravity.

A cylinder with moment of inertia about its center of mass, mass , and radius has a string wrapped around it which istied to the ceiling . The cylinder's vertical position as a function oftime is .

At time the cylinder is released from rest at a height abovethe ground.

Part A

The string constrains the rotational and translational motion of the cylinder. What is the relationship between the angularrotation rate and , the velocity of the center of mass of the cylinder?

Remember that upward motion corresponds to positive linear velocity, and counterclockwise rotation corresponds topositive angular velocity.

Express in terms of and other given quantities.

Hint 1. Key to the constrained motion

Since the cylinder is wrapped about the string, it rolls without slipping with respect to the string. The constraintrelationship to assure this kind of motion can be obtained by considering the relative velocity of the point on thecylinder that is in contact with the string with respect to the stringit should be zero.

Hint 2. Velocity of contact point

Find a general expression for the velocity of the point of contact.

Express your answer in terms of the velocity of the center of the cylinder, , , and therotation rate of the cylinder, .

Hint 1. How to approach this question

First, find the velocity of the contact point relative to the center of the cylinder. Since this ignores thetranslational motion of the entire cylinder with respect to the ground, you then need to add back in the

Chapter 10 [ Edit ]

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I m r

y(t)

t = 0 h

ω v

ω v

v = dy(t)/dt rω

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cylinder's centerofmass velocity to find the velocity of the contact point with respect to the ground.

ANSWER:

ANSWER:

Part B

In similar problems involving rotating bodies, you will often also need the relationship between angular acceleration, ,and linear acceleration, . Find in terms of and .

ANSWER:

Part C

Suppose that at a certain instant the velocity of the cylinder is . What is its total kinetic energy, , at that instant?

Express in terms of , , , and .

Hint 1. Rotational kinetic energy

Find , the kinetic energy of rotation of the cylinder.

Express your answer in terms of and .

ANSWER:

Hint 2. Rotational kinetic energy in terms of

Now, use the results of Part A to express the rotational kinetic energy in terms of , , and .

ANSWER:

Hint 3. Translational kinetic energy

= vcontact

= ω

αa α a r

= α

v Ktotal

Ktotal m r I v

Krot

I ω

= Krot

v

Krot I v r

= Krot



Find , the translational kinetic energy.

Express your answer in terms of and .

ANSWER:

ANSWER:

Part D

Find , the cylinder's vertical velocity when it hits the ground.

Express , in terms of , , , , and .

Hint 1. Initial energy

Find the cylinder's total mechanical energy when the cylinder is released from rest at height . Take thegravitational potential energy to be zero at .

Give your answer in terms of , , and the magnitude of the acceleration due to gravity, .

ANSWER:

Hint 2. Energy conservation

Apply conservation of energy to this situation, to find the total kinetic energy of the cylinder just before it hits theground .

Your answer should involve , and not .

ANSWER:

ANSWER:

Ktrans

m v

= Ktrans

= Ktotal

vf

vf g h I m r

hy = 0

m h g

= Ei

(y = 0)Ktotal

h vf

$K_\rm total(y=0) =

Also accepted:

= vf



Exercise 10.1Description: Calculate the torque (magnitude and direction) about point O due to the force F_vec in each of the casessketched in the figure . In each case, the force F_vec and the rod both lie in the plane of the page, the rod has length 4.00m, and the force has ...

Calculate the torque (magnitude and direction) about point due to the force in each of the cases sketched in the figure .In each case, the force and the rod both lie in the plane of thepage, the rod has length 4.00 , and the force has magnitude 15.0 .

Part A

Calculate the magnitude of the torque in case (a).

ANSWER:

Part B

Find the direction of the torque in case (a).

ANSWER:

Part C

Calculate the magnitude of the torque in case (b)

ANSWER:

O F

F

mN

= = 60.0 |τ| N ⋅ m

into the page out of the page to the right upward the torque is zero



Part D

Find the direction of the torque in case (b).

ANSWER:

Part E

Calculate the magnitude of the torque in case (c)

ANSWER:

Part F

Find the direction of the torque in case (c).

ANSWER:

Part G

Calculate the magnitude of the torque in case (d)

ANSWER:

= = 52.0 |τ| N ⋅ m


= = 30.0 |τ| N ⋅ m


= = 26.0 |τ| N ⋅ m



Part H

Find the direction of the torque in case (d).

ANSWER:

Part I

Calculate the magnitude of the torque in case (e)

ANSWER:

Part J

Find the direction of the torque in case (e).

ANSWER:

Part K

Calculate the magnitude of the torque in case (f)

ANSWER:

Part L

Find the direction of the torque in case (f).

ANSWER:


= 0 |τ| N ⋅ m


= 0 |τ| N ⋅ m



Exercise 10.2Description: (a) Calculate the net torque about point O for the two forces applied as in the figure . The rod and bothforces are in the plane of the page. Take positive torques to be counterclockwise.

Part A

Calculate the net torque about point for the two forces applied as in the figure . The rod and both forces are in theplane of the page. Take positive torques to becounterclockwise.

ANSWER:

Exercise 10.6Description: A metal bar is in the xyplane with one end of the bar at the origin. A force F_vec= ( A )i_unit + ( B )j_unit isapplied to the bar at the point x= x , y= y. (a) What is the position vector r_vec for the point where the force is...

A metal bar is in the xyplane with one end of the bar at the origin. A force 6.50 2.80 is applied to thebar at the point 2.23 , 3.75 .

Part A

What is the position vector for the point where the force is applied?

Enter the x and y components of the radius vector separated by a comma.

ANSWER:


O

= 28.0 τ N ⋅ m

= (F N ) + (i N )jx = m y = m

r



Part B

What are the magnitude of the torque with respect to the origin produced by ?

Express your answer with the appropriate units.

ANSWER:

Part C

What are direction of the torque with respect to the origin produced by ?

ANSWER:

Exercise 10.9Description: The flywheel of an engine has moment of inertia I about its rotation axis. (a) What constant torque isrequired to bring it up to an angular speed of 400 rev/min in 8.00 s, starting from rest?

The flywheel of an engine has moment of inertia 2.70 about its rotation axis.

Part A

What constant torque is required to bring it up to an angular speed of 400 in 8.00 , starting from rest?


ANSWER:

Exercise 10.12

, = , = 2.23, 3.75 rx ry m

F

= = 30.6τ

F

direction direction

direction direction direction

direction

+x

+y

+z

−x

−y

−z

kg ⋅ m2

rev/min s

= = 14.1τ



Description: A stone is suspended from the free end of a wire that is wrapped around the outer rim of a pulley, similar towhat is shown in . The pulley is a uniform disk with mass m and radius r and turns on frictionless bearings. You measurethat...

A stone is suspended from the free end of a wire that is wrapped around the outer rim of a pulley, similar to what is shown in .The pulley is a uniform disk with mass 10.0 and radius 39.0 and turns on frictionlessbearings. You measure that the stone travels a distance 12.3 during a time interval of 2.50 starting from rest.

Part A

Find the mass of the stone.


ANSWER:

Part B

Find the tension in the wire.


ANSWER:

Exercise 10.20Description: A string is wrapped several times around the rim of a small hoop with radius 8.00 cm and mass m. The freeend of the string is held in place and the hoop is released from rest (the figure ). After the hoop has descended s,calculate (a) the angular...

A string is wrapped several times around the rim of a small hoop with radius 8.00 and mass 0.180 . The free end of thestring is held in place and the hoop is released from rest (the figure ). After the hoop has descended 85.0 , calculate

kg cmm s

= = 3.36m

= = 19.7T

cm kgcm



Part A

the angular speed of the rotating hoop and

ANSWER:

Part B

the speed of its center.

ANSWER:

Exercise 10.22Description: A hollow, spherical shell with mass m rolls without slipping down a slope angled at theta. (a) Find theacceleration. (b) Find the friction force. (c) Find the minimum coefficient of friction needed to prevent slipping.

A hollow, spherical shell with mass 2.45 rolls without slipping down a slope angled at 32.0 .

Part A

Find the acceleration.

Take the free fall acceleration to be = 9.80

ANSWER:

= = 36.1 ω rad/s

= = 2.89 v m/s

kg ∘

g m/s2



Part B

Find the friction force.

Take the free fall acceleration to be = 9.80

ANSWER:

Part C

Find the minimum coefficient of friction needed to prevent slipping.

ANSWER:

Exercise 10.29Description: A playground merrygoround has radius R and moment of inertia I about a vertical axle through its center,and it turns with negligible friction. (a) A child applies an F force tangentially to the edge of the merrygoround for t. Ifthe...

A playground merrygoround has radius 2.90 and moment of inertia 2100 about a vertical axle through its center,and it turns with negligible friction.

Part A

A child applies an 21.0 force tangentially to the edge of the merrygoround for 15.0 . If the merrygoround is initiallyat rest, what is its angular speed after this 15.0 interval?

ANSWER:

Part B

How much work did the child do on the merrygoround?

ANSWER:

= 3.12 m/s2

g m/s2

= 5.09 N

= 0.250

m kg ⋅ m2

N ss

= = 0.435 ω rad/s



Part C

What is the average power supplied by the child?

ANSWER:

Exercise 10.31Description: A mkg grinding wheel is in the form of a solid cylinder of radius 0.100 m. (a) What constant torque will bringit from rest to an angular speed of 1200 rev/min in 2.5 s? (b) Through what angle has it turned during that time? (c) Use...

A 2.75 grinding wheel is in the form of a solid cylinder of radius 0.100 .

Part A

What constant torque will bring it from rest to an angular speed of 1200 in 2.5 ?


ANSWER:

Part B

Through what angle has it turned during that time?

ANSWER:

Part C

Use equation to calculate the work done by the torque.


= = 199 W J

= = 13.2 P W

kg m

rev/min s

= = 0.69

Also accepted: = 0.691 , = 0.69

τ

= 160

Also accepted: 157, 160

φ rad

W = ( − ) = Δθτz θ2 θ1 τz



ANSWER:

Part D

What is the grinding wheel’s kinetic energy when it is rotating at 1200 ?


ANSWER:

Part E

Compare your answer in part (D) to the result in part (C).

ANSWER:

Exercise 10.35Description: A x rock has a horizontal velocity of magnitude 12.0 m/s when it is at point P in the figure . (a) At thisinstant, what is the magnitude of its angular momentum relative to point O? (b) What is the direction of the angularmomentum in part (A)? (c)...

A 2.60 rock has a horizontal velocity of magnitude 12.0 when it is at point in the figure .

= = 110

Also accepted: = 109 , = 110

W

rev/min

= = 110


K

The results are the same. The results are not the same.

kg m/s P



Part A

At this instant, what is the magnitude of its angular momentum relative to point ?

ANSWER:

Part B

What is the direction of the angular momentum in part (A)?

ANSWER:

Part C

If the only force acting on the rock is its weight, what is the magnitude of the rate of change of its angular momentum atthis instant?

ANSWER:

Part D

What is the direction of the rate in part (C)?

ANSWER:

O

= = 150 L kg ⋅ /sm2

into the page out of the page

= = 163 ∣∣dL

dt∣∣ kg ⋅ /m2 s2



Exercise 10.36Description: A woman with mass 50 kg is standing on the rim of a large disk that is rotating at 0.80 rev/s about an axisthrough its center. The disk has mass m and radius 4.0 m. (a) Calculate the magnitude of the total angular momentum ofthe...

A woman with mass 50 is standing on the rim of a large disk that is rotating at 0.80 about an axis through its center.The disk has mass 130 and radius 4.0 .

Part A

Calculate the magnitude of the total angular momentum of the woman–disk system. (Assume that you can treat thewoman as a point.)

Express your answer to two significant figures and include the appropriate units.

ANSWER:

Exercise 10.41Description: Under some circumstances, a star can collapse into an extremely dense object made mostly of neutronsand called a neutron star. The density of a neutron star is roughly 10^14 times as great as that of ordinary solid matter.Suppose we represent the...

Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutronstar. The density of a neutron star is roughly times as great as that of ordinary solid matter. Suppose we represent thestar as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was 6.0×105 (comparableto our sun); its final radius is 17 .

Part A

If the original star rotated once in 27 days, find the angular speed of the neutron star.

Express your answer using two significant figures.

ANSWER:

into the page out of the page

kg rev/skg m

= = 9200


L

1014

kmkm

= = 3400 ω2 rad/s



Exercise 10.42Description: A diver comes off a board with arms straight up and legs straight down, giving her a moment of inertia abouther rotation axis of 18 ( kg) * m^2. She then tucks into a small ball, decreasing this moment of inertia to 3.6 ( kg) * m^2.While tucked,...

A diver comes off a board with arms straight up and legs straight down, giving her a moment of inertia about her rotation axisof . She then tucks into a small ball, decreasing this moment of inertia to . While tucked, she makes twocomplete revolutions in 1.3 .

Part A

If she hadn't tucked at all, how many revolutions would she have made in the 1.5 from board to water?

Express your answer using two significant figures.

ANSWER:

Problem 10.62Description: A block with mass m=## kg slides down a surface inclined 36.9 degree(s) to the horizontal (the figure ). Thecoefficient of kinetic friction is ##. A string attached to the block is wrapped around a flywheel on a fixed axis at O. Theflywheel has...

A block with mass = 5.00 slides down a surface inclined 36.9 to the horizontal (the figure ). The coefficient of kineticfriction is 0.27. A string attached to the block is wrapped around aflywheel on a fixed axis at . The flywheel has mass 25.0 andmoment of inertia 0.500 with respect to the axis of rotation.The string pulls without slipping at a perpendicular distance of 0.200 from that axis.

Part A

What is the acceleration of the block down the plane?

ANSWER:

18kg ⋅ m2 3.6kg ⋅ m2

s

s

= 0.46 rev

m kg ∘

O kgkg ⋅ m2

m

= = 1.08 a m/s2



Part B

What is the tension in the string?

ANSWER:

Problem 10.68Description: A thinwalled, hollow spherical shell of mass m and radius r starts from rest and rolls without slipping downthe track shown in the figure . Points A and B are on a circular part of the track having radius R. The diameter of the shellis very...

A thinwalled, hollow spherical shell of mass and radius starts from rest and rolls without slipping down the track shown inthe figure . Points and are on a circular part of the track havingradius . The diameter of the shell is very small compared to and , and the work done by the rolling friction is negligible.

Part A

What is the minimum height for which this shell will make a complete looptheloop on the circular part of the track?

Express your answer in terms of the variables , , and appropriate constants.

ANSWER:

Part B

How hard does the track push on the shell at point , which is at the same level as the center of the circle?


ANSWER:

= = 13.4 T N

m rA B

R h0R

h0

m R

= h0

B

m R



Part C

Suppose that the track had no friction and the shell was released from the same height you found in part (a). Would itmake a complete looptheloop?

ANSWER:

Part D

In part (c), how hard does the track push on the shell at point , the top of the circle?


ANSWER:

Part E

How hard did the track push on the shell at point in part (a)?


ANSWER:

Problem 10.70 CopyDescription: A solid, uniform ball rolls without slipping up a hill, as shown in the figure . At the top of the hill, it is movinghorizontally, and then it goes over the vertical cliff. (a) How far from the foot of the cliff does the ball land? (b) How fastis...

A solid, uniform ball rolls without slipping up a hill, as shown in the figure . At the top of the hill, it is moving horizontally, andthen it goes over the vertical cliff.

= N

h0

yes no

A

m R

= ∣∣N

∣∣

A

m R

= 0∣∣N

∣∣



Part A

How far from the foot of the cliff does the ball land?

ANSWER:

Part B

How fast is it moving just before it lands?

ANSWER:

Problem 10.85Description: A m_bird bird is flying horizontally at v, not paying much attention, when it suddenly flies into a stationaryvertical bar, hitting it h below the top (the figure ). The bar is uniform, l long, has a mass of m, and is hinged at its base.The...

A 550.0 bird is flying horizontally at 2.40 , not paying much attention, when it suddenly flies into a stationary verticalbar, hitting it 25.0 below the top (the figure ). The bar is uniform, 0.740 long, has a mass of 1.30 , and is hinged atits base. The collision stuns the bird so that it just drops to the ground afterward (but soon recovers to fly happily away).

= 36.5 l m

= 28.0 v m/s

−g m/scm m kg



Part A

What is the angular velocity of the bar just after it is hit by the bird?

ANSWER:

Part B

What is the angular velocity of the bar just as it reaches the ground?

ANSWER:

= = 2.73 ω rad/s

= = 6.87 ω rad/s

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Unwinding Cylinder - Physics and Astronomy at TAMU · Unwinding Cylinder Description: Using ... A cylinder with moment of inertia about its center of mass, mass , and radius has a