Calculating Kinetic Energy and Speed of a Rolling Sphere on an Inclined Surface

[Clutch306]

A solid sphere of radius 0.100m and mass 14.3kg rolls without slipping up a surface inclined at 30 degrees to the horizontal. At a certain initial position the sphere's total kinetic energy is 90.1J.
(a) How much of this initial kinetic energy is rational?
(b) What is the speed of the center of mass of the sphere at the initial position?
After the sphere has moved 1.00m up along the incline from its initial position:
(c) What is its total kinetic energy?
(d) What is the speed of its center of mass?

 

[outandbeyond2004]

"rational?" rotational is what I think you meant to write. Well, it would be nice to have some energy to converse with. I do believe I have some energy to help you anyway.

Well, what's the first item you do not understand how to compute, for example, the moment of inertia of the sphere? aah got to go

 

[M98Ranger]

(a) To calculate the rational kinetic energy, we need to first determine the potential energy of the sphere at the initial position. The potential energy is given by the formula PE = mgh, where m is the mass of the sphere, g is the acceleration due to gravity, and h is the height of the sphere above the ground. Since the sphere is at a 30 degree incline, the height can be calculated using trigonometry as h = 0.100m * sin(30) = 0.050m. Plugging in the values, we get PE = (14.3kg)(9.8m/s^2)(0.050m) = 7.01J. Therefore, the rational kinetic energy is 90.1J - 7.01J = 83.09J.

(b) To find the speed of the center of mass at the initial position, we can use the formula for kinetic energy KE = 1/2mv^2, where m is the mass of the sphere and v is the speed of the center of mass. Plugging in the values, we get 90.1J = (1/2)(14.3kg)v^2. Solving for v, we get v = √(2*90.1J/14.3kg) = 2.98m/s. Therefore, the speed of the center of mass at the initial position is 2.98m/s.

(c) To find the total kinetic energy after the sphere has moved 1.00m up the incline, we can use the formula for potential energy PE = mgh, where m is the mass of the sphere, g is the acceleration due to gravity, and h is the height of the sphere above the ground. Since the sphere has moved 1.00m up the incline, the height is now h = 0.100m * sin(30) + 1.00m = 1.05m. Plugging in the values, we get PE = (14.3kg)(9.8m/s^2)(1.05m) = 138.99J. Therefore, the total kinetic energy is 138.99J.

(d) To find the speed of the center of mass after the sphere has moved 1.00m up the incline, we can use the formula for kinetic energy KE = 1/