1. At t=0, a fly wheel has an angular velocity of 4.7 rad/s (radians per second), an angular acceleration of -0.25 rad/s^2 , and a reference line at pheta0 = 0. (a) Through what maximum angle phetamax will the reference line turn in the positive direction ? At what times t will the reference line be at (b) pheta = 1/2phetamax and (c) pheta = -10.5 rad (consider both positive and negative values of t ) ?
2. (a) What is the angular speed w about the polar axis of a point on Earths surface at a latitude of 40 degrees N ? ( earth rotates about that axis ) (b) What is the linear speed v of the point ? What are (c) w and (d) v for a point at the equator ?
3. A record turntable is rotating at 331/3 rev/min. A watermelon seed is on the turntable 6.0 cm from the axis of rotation. (a) Calculate the acceleration of the seed, assuming that it does not slop. (b) What is the minimum value of the coefficient of static friction between the seed and the turntable if the seed is not to slip ? (c) Suppose that the turntable achieves its angular speed by starting from rest and undergoing a constant angular acceleration for 0.25 s. Calculate the minimum coefficient of static friction required for the seed not to slip during the acceleration period.
4. Four identical particles of mass 0.50kg each are placed at the vertices of a 2.0m x 2.0m square and held there by four massless rods , which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) passes through the midpoints of opposite sides and lies in the plane of the square, (b) passes through the midpoint of one of the sides and is perpendicular to the plane of the square , and (c) lies in the plane of the square and passes through two diagonally opposite particles ?
5. Delivery trucks that operate by making use of energy stored in a rotating flywheel have been used in Europe. The trucks are charged by using an electric motor to get the flywheel up to its top speed of 200 pie radians/second. One such flywheel is a solid uniform cylinder with a mass of 500kg and a radius of 1.0m (a) What is the kinetic energy of the flywheel after charging ? (b) If the truck operates with an average power requirement of 8.0 kW, for how many minutes can it operate between charging?
6. A door has a mass of 44,000 kg , a rotational inertia about a vertical axis throught its huge hinges of 8.7 x 10^4 kg * m^2 , and a (front) face width of 2.4 m. Neglecting friction , what steady force applied at its outer edge and perpendicular to the plane of the door, can move it from rest through an angle of 90 degree's in 30 seconds ?
7. Figure shows a rigid assembly of thin hoop (of mass m and radius R = 0.150m ) and a thin radial rod ( of mass m and length L = 2.00 R ). The assembly is upright but if we give it a slight push it will rotate around a horizontal axis in the plane of the rod and hoop., through the lower end of the rod. Assuming that the energy given to the assembly in such a push is negligible, what would be the assembly's angular speed about the rotation axis when it passes through the upside-down (inverted) orientation ?
In order to calculate the angular speed, we need to determine the amount of energy that the object gains. It gains energy due to gravity. The easiest way of doing this is to compare the final and initial positions, and then find their difference in energy. We calculate the energy by knowing the center of mass of the object, and the difference in heights of the center of mass.
First, we calculate the center of mass. The center of mass for the ring by itself is directly in the center. It is therefore a distance R above the end of the rod. Since the rod is length 2*R, the center of mass of the ring is 3*R above the plane.
The center of mass of the rod is directly in its ...
With clear explanations and calculations, the problems are solved.