Question

1) The parallel axis theorem provides a useful way to calculate the moment of inertia I about an arbitrary axis. The theorem states that I = Icm + Mh2, where Icm is the moment of inertia of the object relative to an axis that passes through the center of mass and is parallel to the axis of interest, M is the total mass of the object, and h is the perpendicular distance between the two axes. Use this theorem and information to determine the moment of inertia (kg·m2) of a solid cylinder of mass M = 2.50 kg and radius R = 2.50 m relative to an axis that lies on the surface of the cylinder and is perpendicular to the circular ends.

2) A flywheel is a solid disk that rotates about an axis that is perpendicular to the disk at its center. Rotating flywheels provide a means for storing energy in the form of rotational kinetic energy and are being considered as a possible alternative to batteries in electric cars. The gasoline burned in a 231-mile trip in a typical midsize car produces about 2.97 x 10⁹ J of energy. How fast would a 45.3-kg flywheel with a radius of 0.209 m have to rotate to store this much energy? Give your answer in rev/min.

3) Starting from rest, a basketball rolls from the top to the bottom of a hill, reaching a translational speed of 6.5 m/s. Ignore frictional losses. (a) What is the height of the hill? (b) Released from rest at the same height, a can of frozen juice rolls to the bottom of the same hill. What is the translational speed of the frozen juice can when it reaches the bottom?

4) A playground carousel is rotating counterclockwise about its center on frictionless bearings. A person standing still on the ground grabs onto one of the bars on the carousel very close to its outer edge and climbs aboard. Thus, this person begins with an angular speed of zero and ends up with a nonzero angular speed, which means that he underwent a counterclockwise angular acceleration. The carousel has a radius of 1.69 m, an initial angular speed of 3.22 rad/s, and a moment of inertia of 121 kg·m². The mass of the person is 45.0 kg. Find the final angular speed of the carousel after the person climbs aboard.

Ive tried figuring these out but i need help.

Answer 1

1.
Since cylinder is solid,
Icm = 0.5*M*r^2
= 0.5*2.5* (2.5)^2
= 7.81 Kgm^2

h = 2.5 cm here

I = Icm + M*h^2
= 7.81 + 2.5*(2.5)^2
= 122.0 Kg.m^2

2.
Moment of inertia of solid disk, I = 0.5*m*r^2
= 0.5*45.3*(0.209)^2
= 0.989 Kg.m^2

rotational kinetic energy= 0.5*I*w^2
2.97*10^9 = 0.5*0.989*w^2
w= 7.75*10^4 rad/s

1s = 1/60 min
1 rad = 1/2pi rev
so,
rad/s = 60/2*pi rev/min =9.55 rev/min

so,
w= 7.75*10^4 rad/s
=7.75*10^4 *9.55 rev/min
= 7.4*10^-3 rev/min

only 1 question at a time please. I have answered 2