A group of students perform the same "Conservation of Mechanical Energy" experiment that you performed in lab by allowing a solid sphere and then a solid cylinder to roll down the ramp. The solid sphere was released from a height of 12.9 cm. From what height hcylinder should the solid cylinder be released so that it has the same speed as the solid sphere when it reaches the bottom of the ramp?
Solid Sphere
KE1 + PE1 = KE2 + PE2
0 + m g h = 0.5 m v^2 + 0.5 I w^2 + 0
I = 0.4 m r^2
m g h = 0.5 m v^2 + 0.5 * 0.4 m r^2 (v/r)^2
m g h = 0.5 m v^2 + 0.2 m v^2
g h = 0.7 v^2
v = sqrt(gh/0.7) = sqrt( 9.81 x 0.129 / 0.7) = 1.345 m/s
Solid Cylinder
KE1 + PE1 = KE2 + PE2
0 + m g h = 0.5 m v^2 + 0.5 I w^2 + 0
I = 0.5 m r^2
m g h = 0.5 m v^2 + 0.5 * 0.5 m r^2 (v/r)^2
m g h = 0.5 m v^2 + 0.25 m v^2
g h = 0.75 v^2
h = 0.75 v^2 / g = 0.75 x (1.345)^2 / 9.81 = 0.1385 m = 13.85 cm
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Height = 13.85 cm
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