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Practice problems with rotational inertia - 7 Problems

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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 ?

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A set of 7 problems which can serve a good set of practice problems. Answer is given with all steps and mathematical calculations. A MS Word attachment is given.

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1. At t = 0, a flywheel has an angular velocity of 4.7 rad/s, an angular acceleration of -0.25 rad/s2, and a reference line at . (a) Through what maximum angle will the reference line turn in the positive direction? At what times will the line be at (b) and (c) (consider both positive and negative values of )?
Solution
The basic equations for angular motion are

and

(a) The maximum positive angle will occur when . From equation (2) we see that this occurs when . For our problem so
t = 18.8s. Putting this value of t into equation (1) gives .

(b) Use in equation (1) and solve for t. Doing so gives
. The second solution comes about because the wheel reaches and then turns back, turning through .

(c) Use in equation (1) and solve for t. Doing so gives
.

2 (a) what is the angular speed w about the polar axis of a point on Earths surface at 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?

It takes the Earth approximately 23 hours, 56 minutes and 4.09 seconds to make one complete revolution (360 degrees). This length of time is known as a sidereal day. In comparison to some other planets such as Pluto and Mercury, the Earth rotates at a very slow angular velocity of approximately 7.272205217 x 10-5 radians/second.
In the non-rotating frame, the Earth rotates below a pendulum at the North Pole with angular speed w = 2&#61552;/P, where P = sidereal day. At latitude f, the vertical component of the Earth's angular speed is w' = w sin f
a) Here f = 40, thus w` = w Sin40 = 7.272205217 x 10-5 radians/second. * Sin40
b) Linear velocity and angular velocity are related as, Vlin = w'*R, R is the radius of earth

c) At the equator, f = 0 hence w = 0

3. A record turntable is rotating at 33 1/3 rev/min. A watermelon seed is on the turntable 6.0 ...

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