In a common lecture demonstration, an instructor \"races\" various round objects
ID: 1352973 • Letter: I
Question
In a common lecture demonstration, an instructor "races" various round objects by releasing them from rest at the top of an inclined plane and letting them roll down the plane (Figure 1) . Before the objects are released, the students guess which object will win. How much faster is the solid cylinder moving at the bottom of the ramp than the hollow cylinder?
a.) Find the ratio of the final speeds for a solid sphere and a solid cylinder expressed in. vcm,1/vcm,2=?
b.) What is the ratio of the rotational kinetic energy of the solid sphere to the solid cylinder at the bottom of the ramp? Krot,1Krot,2=?
Explanation / Answer
by conservation of energy
initial energy = final energy
mgh = rotatioal kinetic energy + translational kinetic energy
for sphere
mgh = 0.5 * (2/5) * m * r^2 * Vcm1^2 / r^2 + 0.5 * m * vcm1^2
gh = 0.5 * (2/5) vcm1^2 + 0.5 * vcm1^2
vcm1^2 = gh / ( 0.5 * (2/5) + 0.5)
for solid cylinder
mgh = 0.5 * 0.5 * m * r^2 * vcm1^2 / r^2 + 0.5 * m * vcm1^2
gh = 0.5 * 0.5 * vcm2^2 + 0.5 * vcm2^2
vcm2^2 = gh / (0.5 * 0.5 + 0.5)
vcm1/vcm2 = sqrt((0.5 * 0.5 + 0.5) / (0.5 * (2/5) + 0.5))
vcm1/vcm2 = 1.035098
Krot1 / Krot2 = 0.5 * (2/5) vcm1^2 / 0.5 * 0.5 * vcm2^2
Krot1 / Krot2 = 0.8 * (vcm1 / vcm2)^2
Krot1 / Krot2 = 0.8 * 1.035098^2
Krot1 / Krot2 = 0.857