A gyroscope consists of a flywheel of mass m, which has a moment of inertia I for rotation about its axis. It is mounted on a rod of negligible mass, which is supported at one end by a frictionless pivot attached to a vertical post, as shown in the diagram. The distance between the center of the wheel and the pivot is d. The wheel rotates about its axis with angular velocity omega, where positive omega refers to counterclockwise rotation as seen by an observer looking at the face of the wheel that is opposite the pivot. The rod is tilted upward, making an angle theta with respect to the horizontal. Gravity acts downward with a force of magnitude m g.

Adopt a coordinate system with the z axis pointing upward and the x and y axes in the horizontal plane. The gyroscope is moving, but at t = 0, the rod is in the yz plane.

a). Assuming that the only significant contribution to the angular momentum comes from the spinning of the flywheel about its center, what is the angular momentumvector Lvec about the pivot at t = 0?

Specify the components of Lvec with respect to the axes shown in the diagram. Write the components in order Lx, Ly, Lz separated by commas.

b). At t = 0, what is the torque acting on the wheel about the pivot? (Express your answer in terms of x,y and z components)

c). The gyroscope is observed to precess about the vertical axis, with an angular velocity of precession Omega, defined as positive for counterclockwise precession as seen from above. Find Omega in terms of the given quantities.

1. A torque is a product of a force and a distance (lever arm). Work is also the product of a force and a distance. Yet, "torque" and "work" are different. What is it about the distances that make "torque" and "work" different?
2. What is a gyroscope? What does the angular momentum has to do with a gyroscope? Provide examples

Air is trapped in the lower part of a thin tube below a mercury column of length 8 cm. When the tube is vertical the length of air column trapped is 12 cm. The top is open and the system is at equilibrium. What will be the length of the trapped air column if the tube is now tilted so that it makes an angle of 65 degrees to the v

You stand 1.66 m in front of a wall and gaze downward at a small vertical mirror mounted on it. In this mirror you can see the reflection of your shoes. If your eyes are 1.74 m above your feet, through what angle should the mirror be tilted for you to see your eyes reflected in the mirror? (The location of the mirror remains the

1. If the axis of the Earth were tilted at 50 instead of 23.5, would the Summers in Europe be longer and hotter ? Explain.
2. Why do we always see the same face of the Moon from the Earth?
3. If a lunar eclipse occurs in January, can it be seen from the South Pole? Explain.

1. Explain the difference between linear speed and rotational speed. Include formula difference and how rotational speed is changed to linear speed.
2. A bicycle has wheels with a 30-centimeter radius. If smaller wheels with a 25-centimeter radius are installed, what is the change in linear speed of the bicycle if the wheels

A block lies on a plane tilted at an angle theta = 22.0* to the horizontal, with the coefficient of kinetic friction mu_k = 0.17.
a) Determine the acceleration of the block as it slides down the plane.
b) If the block starts from rest 9.3 m up the plane from its base, what will be the block's speed when it reaches the bottom o

A plane mirror is placed in front of a multiple ray projector so that the central ray is back upon itself. The other rays are adjusted so they are also reflected back on themselves. In this way, all 5 rays are parallel.
A concave mirror is placed in the path of the rays so that the central ray is again reflected back upon

A meterstick is parallel to the x axis in S and is moving in the +y direction at constant speed vy. Show that the meterstick will appear tilted at an angle theta' with respect to the x'axis of S' moving in the +x direction at beta=0.65. Compute the angle theta' measured in S'.

5. A rectangular block twice as high as it is wide is resting on a board. The coefficient of static friction between board and incline is .63. If the board is tilted will the block first tip over or begin sliding?
6. A uniform pole of mass M is at rest on an incline of angle secured by a horizontal rope. For what angle