A marble and a cube are placed at the top of a ramp. Starting from rest at the same height, the marble rolls without slipping and the cube slides (no kinetic friction) down the ramp. Determine the ratio of the center-of-mass speed of the cube to the center-of-mass speed of the marble at the bottom of the ramp.
To determine the ratio of the center-of-mass (CM) speed of the cube to the CM speed of the marble at the bottom of the ramp, we can apply the principles of rotational and translational motion for each object.
Let's start by considering the marble. Since it rolls without slipping, it undergoes both rotational and translational motion. The relationship between these motions can be described by the equation:
v = ω * r
Where:
v is the translational speed of the marble's CM,
ω is the angular speed (rotational speed) of the marble,
r is the radius of the marble.
For the cube, it only undergoes translational motion since there is no kinetic friction acting on it. The translational speed of the cube's CM can be determined using the Newton's second law of motion:
F = m * a
Since the only force acting on the cube is gravity, we can write:
mg * sinθ = m * a
Where:
m is the mass of the cube,
g is the acceleration due to gravity,
θ is the angle of the ramp.
The acceleration of the cube can be related to its translational speed using the equation:
a = v' / t
Where:
v' is the translational speed of the cube's CM,
t is the time taken to reach the bottom of the ramp.
Now, let's relate the translational speed of the cube to the angular speed of the marble:
v' = ω' * r'
Where:
v' is the translational speed of the cube's CM,
ω' is the angular speed of the imaginary sphere with the same radius as the cube,
r' is the distance of the cube's CM from the rotation axis (center of the ramp).
Since both objects start from rest at the same height, we can assume that they both reach the bottom of the ramp at the same time (t). Therefore, we can set up the following ratio:
(v' / v) = (ω' * r') / (ω * r)
Now, we need to relate the angular speeds of the marble and the cube. Since the ramp is not specified, we can assume that they both have the same angle (θ). The ratio between their angular speeds can be given by:
ω' / ω = (r' / r)
Substituting this ratio into our previous expression:
(v' / v) = (ω' * r') / (ω * r) = (r' / r) = 1
Therefore, the ratio of the CM speed of the cube to the CM speed of the marble at the bottom of the ramp is 1. This means that both objects have the same center-of-mass speed at the bottom of the ramp.