Chapter 11: Angular Momentum

Because the Earth is nearly perfectly spherical, gravitational forces act on it effectively through its center. Explain why the Earth’s axis points at the North star all year long. Also explain why the earth’s rotation speed stays the same throughout the year (one rotation per $\mathbf{24}\mathit{h}$). In your analysis, does it matter that the Earth is going around the sun?

In actual fact, the Earth is not perfectly spherical. It bulges out a bit at the equator, and tides tend to pile up water at one side of the ocean. As a result, there are small torque exerted on the Earth by other bodies, mainly the sun and the moon. Over many thousands of years there are changes in what portion of sky the Earth’s axis points towards (Change of direction of rotational angular momentum), and changes in the length of a day (change of magnitude of rotational angular momentum).

Make a sketch showing a situation in which the torque due to a single force about some location is $20\mathrm{N}\cdot \mathrm{m}$ in the positive $z$ direction, whereas about another location the torque is $10\mathrm{N}\cdot \mathrm{m}$ in the negative $z$ direction.

Two gyroscopes are made exactly alike except that the spinning disk in one is made of low-density aluminum, whereas the disk in the other is made of high-density lead. If they have the same spin angular speeds and the same torque is applied to both, which gyroscope processes faster?

The axis of gyroscope is tilted at an angle of ${30}^{\circ }$to the vertical (figure 11.110). The rotor has a radius of 15cm , mass 3kg , moment of inertia is $0.06Kg/m^2$and spins on its axis at $30rad/s$. It is supported in a cage (not shown) in such a way that without an added weigh it does not process. Then a mass of 0.2 kg is hung from the axis at a distance of 18 cm from the center of the rotor.

A bicycle wheel with a heavy rim is mounted on a lightweight axle, and one end of the axle rests on top of a post. The wheel is observed to presses in the horizontal plane. With the spin direction shown in figure11.111, does the wheel process clockwise or counter wise? Explain in detail, including appropriate diagrams.

A solid wood top spins at high speed on the floor, with a spin direction shown in figure 11.112

a. Using appropriately labeled diagrams, explain the direction of motion of the top (you do not need to explain the magnitude).

b. How would the motion change if the top had a higher spin rate? Explain briefly.

c. If the top were made of solid steel instead of wood, explain how this would affect the motion (for the same spin rate).

Redo the analysis, calculating torque and angular momentum relative to a fixed location in the ice anywhere underneath the string (similar to the analysis of the meter stick around one end). Show that the two analyses of the puck are consistent with each other.

What is required for the angular momentum of a system to be constant? (a) zero net torque, (b) zero impulse, (c) no energy transfers, (d) zero net force

Model the motion of a meter stick suspended from one end on a low-friction. Do not make the small-angle approximation but allow the meter stick to swing with large angels. Plot on the game graph both$\theta$and the $z$component of $\overrightarrow{\omega }$vs. time, Try starting from rest at various initial angles, including nearly straight up (Which would be ${\theta }_i=\pi$ radians). Is this a harmonic oscillator? Is it a harmonic oscillator for small angles?

Model the motion of a meter stick suspended from one end on a low-friction. Do not make the small-angle approximation but allow the meter stick to swing with large angels. Plot on the game graph both $\theta$and the z component of $\overrightarrow{\omega }$vs. time, Try starting from rest at various initial angles, including nearly straight up (Which would be ${\theta }_i=\pi$ radians. Is this a harmonic oscillator? Is it a harmonic oscillator for small angles?