Hollow Ground
Which would reach the ground first: hollow or solid cylinder?
This is a 4th year student. I need help for Physics.
If a hollow and a solid cylinder were rolled down an inclined plane, which of the two would reach the ground first? Please explain why.
The two have the same mass. Disregard air resistance.
The answer from Balls was really close. It is about moment of inertia but not only that. The solid cylinder actually has a higher moment of inertia than the hollow one. It is the ratio of moment of inertia to mass that determines the speed. The lower the ratio, the faster it rolls. A low ratio means that the mass is more concentrated to the center.
Perhaps the easiest way to explain is to examine the problem in terms of energy. There are three components of energy here:
1. Potential energy: P = m * g * h
2. Linear kinetic energy: L = (m * v^2 )/2
3. Rotational kinetic energy: R = (I * w^2)/2
Where m is the object mass, g is gravity, h is the object height, v is the linear velocity, I is the moment of inertia and w is the angular velocity. Since the object is rolling and is of radius r, there is a fixed relation between velocity and angular velocity:
v = r * w
The moment of inertia of a cylinder is defined as: I = (r^2 + ri^2)*m
Where ri is trhe inner radius (zero for a solid cylinder) The larger mass of the solid cylinder gives it a larger moment of inertia.
The energy of the system is conserved so (arbitrarily assigning the initial energy as 0)
0 = m * g * h + (m * v^2)/2 + ((r^2 + ri^2)*m * (v/r)^2)/2
Simplifying:
0 = g * h + *v^2 *(1 + (ri/r)^2)
Solving for the cylinder velocity:
v = sqrt(g * h / (1 + (ri/r)^2))
Notice that the larger the value of ri, the slower the velocity. ri is limited to be less than r. As r approaches ri, the velocity approaches 1/sqrt(2) of the solid cylinder.
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