Consider the following objects of mass m rolling down an incline of height h. A
ID: 1421848 • Letter: C
Question
Consider the following objects of mass m rolling down an incline of height h. A hoop has a moment of inertia I = mr^2. What is the equation for the velocity V_hoop of the hoop at the bottom of the incline? (Use the following as necessary: m, h, r, and g.) V_hoop = A solid cylinder has a moment of inertia I = 1/2 mr^2. What is the equation for the velocity V_cylinder of the cylinder at the bottom of the incline? (Use the following as necessary: m, h, r, and g.) V_cylinder = We know that the velocity of the sphere at the bottom of the ramp is squareroot 10gh/7 from which we can conclude that the mass of the sphere does not affect the velocity of the sphere. Which of the following statements help to explain why the equations for the velocity in the case of the rolling cylinder and rolling hoop should be different from each other and from that of the sphere? (Select all that apply.) The rotational kinetic energy of a solid depends on its moment of inertia I, which will affect the velocity of the object as it rolls down the incline. Even if all three objects had the same mass and the same radius, they would all have different moments of inertia and therefore different rotational kinetic energies, which will affect the velocity of the object as it rolls down the incline. The moment of inertial of a sphere, cylinder and hoop are different because of how the mass is distributed in each of these objects. This would affect the velocity of the object as it rolls down the incline.Explanation / Answer
a)
Using Energy Conservation,
P.E = K.E + Rotational K.E
m *g* h = 1/2 *m * v^2 + 1/2 * I * w^2
mgh = 1/2 mv^2 + 1/2 mr^2 . (v/r)^2
mgh = mv^2
v = gh
b)
Using Energy Conservation,
P.E = K.E + Rotational K.E
m *g* h = 1/2 *m * v^2 + 1/2 * I * w^2
mgh = 1/2 mv^2 + 1/2 (1/2) *mr^2 . (v/r)^2
mgh = 1/2 mv^2 + 1/4 mv^2
v = (4gh / 3 )
(c)
All three statements are true.