Need Help w/ Some Rotational Physics

[r34racer01]

A student sits on a freely turning stool and rotates with constant angular velocity w1. She pulls her arms in towards her body, and her angular velocity increases to w2.
In doing this her kinetic energy: increases, stays the same or decreases

I thought that since KE = 1/2Iw^2 that would mean an increase in w would be an increse in KE, but I hear otherwise, can someone clarify this for me?
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A boy of mass m = 50 kg running with speed v = 1 m/s jumps onto the outer edge of a merry-go-round of mass M = 100 kg and radius R = 2 m, as shown in the picture above. The merry-go-round is initially at rest, and can rotate about a frictionless pivot at its center. You may assume that the inital velocity of the boy is tangent to the edge of the merry-go round.

Which of the following quantities are conserved throughout this problem for the system consisting of the boy and the merry-go-round?

only kinetic energy
kinetic energy and angular momentum
only linear momentum
linear momentum and angular momentum
only angular momentum

I think linear & angular momentum are conserved because they are both conserved whether the collision is elastic or inelastic right?
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A gyroscope is turning such that in the picture the top of the wheel is coming out of the page and the bottom of the wheel is going into the page (the wheel is rotating in the sense of the curved arrow). The bar (which is attached to the center of mass of the wheel and of negligible mass) is being held up by a piece of string wrapped around its end.

Which direction does the angular momentum vector of the gyroscope point in?

+x
-x
+zThe direction of rotation (precession) of the gyroscope is:

counterclockwise in the plane of the page (rotating in the direction from the -x axis to the -y axis)
out of the page (rotating in the direction from the -x axis to the z axis)
into the page (rotating in the direction from the -x axis to the -z axis)

I am just completely lost on these 2. Can anyone break this down into simpler words?

Homework Statement

Homework Equations

The Attempt at a Solution

 

[Hootenanny]

A student sits on a freely turning stool and rotates with constant angular velocity w1. She pulls her arms in towards her body, and her angular velocity increases to w2.
In doing this her kinetic energy: increases, stays the same or decreases

I thought that since KE = 1/2Iw^2 that would mean an increase in w would be an increse in KE, but I hear otherwise, can someone clarify this for me?

Are you sure that rotational kinetic energy is conserved? Furthermore, are you sure that the moment of inertia remains constant?

A boy of mass m = 50 kg running with speed v = 1 m/s jumps onto the outer edge of a merry-go-round of mass M = 100 kg and radius R = 2 m, as shown in the picture above. The merry-go-round is initially at rest, and can rotate about a frictionless pivot at its center. You may assume that the inital velocity of the boy is tangent to the edge of the merry-go round.

Which of the following quantities are conserved throughout this problem for the system consisting of the boy and the merry-go-round?

only kinetic energy
kinetic energy and angular momentum
only linear momentum
linear momentum and angular momentum
only angular momentum

I think linear & angular momentum are conserved because they are both conserved whether the collision is elastic or inelastic right?

Is the linear momentum really conserved? What is the condition that linear momentum be conserved? Secondly, are there any external torques acting on the system?

A gyroscope is turning such that in the picture the top of the wheel is coming out of the page and the bottom of the wheel is going into the page (the wheel is rotating in the sense of the curved arrow). The bar (which is attached to the center of mass of the wheel and of negligible mass) is being held up by a piece of string wrapped around its end.

Which direction does the angular momentum vector of the gyroscope point in?

+x
-x
+z


The direction of rotation (precession) of the gyroscope is:

counterclockwise in the plane of the page (rotating in the direction from the -x axis to the -y axis)
out of the page (rotating in the direction from the -x axis to the z axis)
into the page (rotating in the direction from the -x axis to the -z axis)

I am just completely lost on these 2. Can anyone break this down into simpler words?

Could you perhaps post the picture?

 

[r34racer01]

Are you sure that rotational kinetic energy is conserved?
Could you perhaps post the picture?

I've added it to the original post and thanks so far for the help.

 

[Cleonis]

A student sits on a freely turning stool and rotates with constant angular velocity w1. She pulls her arms in towards her body, and her angular velocity increases to w2.
In doing this her kinetic energy: increases, stays the same or decreases

I thought that since KE = 1/2Iw^2 that would mean an increase in w would be an increse in KE, but I hear otherwise, can someone clarify this for me?

There is the following comparison:
The motion of the hands of the student is like the motion of a celestial body orbiting the sun, on a non-circular trajectory.

As an extreme example, think of a comet with a very elliptical orbit. Look at the part of the trajectory where the comet is moving towards the center of the solar system. Gravity from the Sun is the centripetal force. The centripetal force is increasing the comet's velocity. This shows that close to the Sun the comet has more kinetic energy than when it is far away from the Sun.

The student rotating away on the free turning stool is providing the centripetal force for her arms. To tuck her arms in she provides a surplus of centripetal force. Therefore when her arms are tucked in her kinetic energy must be larger then before.

As you state, the rotational kinetic energy is 1/2*I*w^2
To compare before and after, you have to compare the moment of inertia before and after, plus the angular velocity before and after.

The comparison with the case of ellipse-shaped orbit provides a check. If your computation appears to show that kinetic energy remains the same then you know there must be an error somewhere; correct computation will show that kinetic energy increases when the student tucks in her arms.

It won't catch all mistakes, though. If you make two errors then you may end up with the correct prediction, reached with wrong reasoning. Your initial attempt does indicate you're making two errors.

 

[rdl]

1.

When the student pulls her arms in towards her body, her angular velocity increases to w2. This is because of the conservation of angular momentum, which states that the total angular momentum of a system remains constant unless acted upon by an external torque. In this case, as the student brings her arms closer to her body, she decreases her moment of inertia (I) which results in an increase in angular velocity (w) to maintain the same angular momentum (L = Iw). Therefore, her kinetic energy increases as well, as KE = 1/2Iw^2.

2.

Linear momentum and angular momentum are both conserved in this problem. Linear momentum is conserved because there are no external forces acting on the system in the horizontal direction. Angular momentum is also conserved because there are no external torques acting on the system. The force of the boy jumping onto the merry-go-round is internal to the system and does not contribute to the change in angular momentum.

3.

The angular momentum vector of the gyroscope points in the +z direction. This is because the gyroscope is rotating in the counterclockwise direction, as indicated by the curved arrow, and the right-hand rule states that the direction of the angular momentum vector is in the direction of rotation.

The direction of rotation (precession) of the gyroscope is out of the page, or in the direction from the -x axis to the z axis. This is because the gyroscope is rotating in the counterclockwise direction and the precession is in the same direction as the angular momentum vector, which points in the +z direction.