Concept #1: Intro to Rotational Kinetic Energy

Practice: A flywheel is a rotating disc used to store energy. What is the maximum energy you can store on a flywheel built as a solid disc with mass 8 x 10^{4} kg and diameter 5.0 m, if it can spin at a max of 120 RPM?

Example #1: Mass of re-designed flywheel

Practice: When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 10,000 J in kinetic energy. Calculate the sphere’s mass.

A solid sphere rolls along a horizontal, smooth surface at a constant linear speed without slipping.
What is the ratio between the rotational kinetic energy about the center of the sphere and the sphere’s total kinetic energy?
A. 3/7
B. None of these
C. 7/2
D. 2/5
E. 2/7
F. 3/5
G. 5/3

A hollow spherical shell has mass 8.20 kg and radius 0.225 m . It is initially at rest and then rotates about a stationary axis that lies along a diameter with a constant acceleration of 0.895 rad/s2 . What is the kinetic energy of the shell after it has turned through 6.25 rev ?

You are designing a rotating metal flywheel that will be used to store energy. The flywheel is to be a uniform disk with radius 31.0 cm . Starting from rest at t = 0, the flywheel rotates with constant angular acceleration 3.00 rad/s2 about an axis perpendicular to the flywheel at its center. If the flywheel has a density (mass per unit volume) of 8600 kg/m3, what thickness must it have to store 800 J of kinetic energy at t = 8.00 s?

A merry-go-round has a mass of 1550 kg and a radius of 7.60 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 9.00 s ? Assume it is a solid cylinder.

Suppose that some time in the future we decide to tap the moons rotational energy for use on earth. In additional to the astronomical data in Appendix F in the textbook, you may need to know that the moon spins on its axis once every 27.3 days. Assume that the moon is uniform throughout.a) How much total energy could we get from the moons rotation?b) The world presently uses about 4.0
1020 J of energy per year. If in the future the world uses five times as much energy yearly, for how many years would the moons rotation provide us energy?c) In light of your answer, does this seem like a cost-effective energy source in which to invest?

The flywheel of a gasoline engine is required to give up 600 J of kinetic energy while its angular velocity decreases from 780 rev/min to 510 rev/min. What moment of inertia is required?

A square, with a side length of 40 cm, has a 5 kg mass at each of its corners. If the square rotates about an axis through its center, perpendicular to the plane of the square, with an angular speed of 5 rad/s, how much kinetic energy does the square have?

If we multiply all design dimensions of an object by a scaling factor f, its volume and mass will be multiplied by f3.a) By what factor will its moment of inertia be multiplied?b) If a (1/48)large{{frac{1}{48}}}-scale model has a rotational kinetic energy of 2.5 J, what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

Biomedical measurements show that the arms and hands together typically make up 13.0 % of a persons mass, while the legs and feet together account for 37.0 % . For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. Let us consider a 76.0 kg person having arms 68.0 cm long and legs 93.0 cm long. The person is running at 12.0 km/h , with his arms and legs each swinging through 30° in 1/2 slarge{frac{1}{2};{
m s}}. Assume that the arms and legs are kept straight.a) What is the average angular velocity of his arms and legs?b) Calculate the amount of rotational kinetic energy in this persons arms and legs as he walks.c) What is the total kinetic energy due to both his forward motion and his rotation?d) What percentage of his kinetic energy is due to the rotation of his legs and arms?

A grinding wheel in the shape of a solid disk is 0.200 m in diameter and has a mass of 3.0 kg. The wheel is rotating at 2200 rpm about an axis through its center.
a) What is its kinetic energy?
b) How far would it have to drop in free fall to acquire the same amount of kinetic energy?

The three 210 g masses in the figure are connected by massless, rigid rods.a) What is the triangle’s moment of
inertia about the axis through the
center?b) What is the triangle’s kinetic energy if it rotates about the axis at 5.2 rev/s ?

The L-shaped object shown in the figure below right consists of the masses connected by light rods. How much work must be done to accelerate the object from rest to an angular speed of 3.25 rad/s about the x-axis?
a) 62.3 J
b) 23.7 J
c) 17.6 J
d) 47.4 J
e) 34.2 J