Sphere Rolling Down a Plane

A hoop of mass M=2 kg, and radius R=0.05 m, is rolling down an inclined plane of vertical height 1 meter, at an angle of 10° above the floor. At the same time, a hollow sphere with the same mass and radius starts rolling down the hill! Assume there is no friction between the objects and the inclined plane. (Please show work)

(1) Find the initial potential energy of the hollow sphere.

(2) Find the initial potential energy of the hoop.

(3) Find the final velocity of the hollow sphere.

(4) Find the final velocity of the hoop.

(5) Find the final translational kinetic energy of the hollow sphere. Is this equal to the initial potential energy? Why or why not?

(6) Which object gets to the bottom of the inclined plane faster? Explain.

Answers

1), 2)

Initially the hoop and the sphere are at the same height. Thus they have the same potential energy.

$Ep= M*g*H =2*9.81*1 =19.62 J$

3, 4)

At the base of inclined all the potential energy is transformed into kinetic energy of rotation + kinetic energy of translation.

$Ep = I*\omega^2/2 + Mv^2/2$

with $v = \omega*R$

$Ep = I*\omega^2/2 + M*\omega^2*R^2/2$

for the hoop $I1 =MR^2$ for the hollow sphere $I2 =(2/3)MR^2$

for hoop:

$Ep =M*\omega^2*R^2$

$\omega1 = \sqrt{Ep/M/R^2} = \sqrt(19.62/2/0.05^2)= 62.64 rad/s$

$v1 = \omega1*R =62.64*0.05 =3.132 m/s$

for the sphere

$Ep = (2/6)M*\omega^2*R^2 +(1/2)*M*\omega^2*R^2 =(5/6)*M*\omega^2*R^2$

$\omega_2 = \sqrt{(6/5)*Ep/M/R^2} =\sqrt {(6/5)*19.62/2/0.05^2} =68.62 rad/s$

$v2 = omega2*R =68.62*0,05 =3.431 m/s$

5) $Ek2 = Mv2^2/2 = 2*3.431^2/2 =11.77 J$

This translational kinetic energy is smaller than the initial potential energy. This is because there is also rotational kinetic energy at the bottom of the incline.

6) The sphere gets to the bottom of the incline faster. This is because it rotates faster, and this happens because it has a smaller inertia moment compared with the hook. (In other words the angular acceleration of the sphere is bigger because of its smaller inertia moment, and thus it begings to rotate faster).