A 1.0- kg mass is attached to a string wrapped around a shaft of negligible mass
ID: 2255135 • Letter: A
Question
A 1.0- kg mass is attached to a string wrapped around a shaft of negligible mass and having a 5.0- cm radius. A dumbbell-shaped "flywheel" made from two 0.500- kg masses is attached to one end of the shaft and perpendicular to its axis (see the figure). The mass is released from rest and allowed to fall 0.8 m to the floor. It reaches a speed of 1.3325 m/s just before striking the floor. How far apart are the masses of the dumbbell?
Explanation / Answer
m=1 kg is the mass of the weight
mF = 0.5 kg is the mass of each flywheel component
RF is the distance of a flywheel weight to the center of rotation
LF is the distance sought between the flywheel weight
R = 0.05 m is the radius of teh shaft
? = 8s is the time it takes for the weight to drom by h = 0.8 m
z(t) is the altitude of the weight, v(t) its speed
z(0) = h, v(?) = 1.4222 m/s
?(t) is the angle of the shaft with its initial position
?(t) is the angular velocity of the shaft
J is the moment of inertia of the shaft
? is the torque applied by the string onto the shaft
T is the tension of the string
We need to solve jointly the equation of motion for the weight (linear translation) and the shaft (rotation), the unkown linking the two being the tension of the string.
The string in inelastic, so the movement of the shaft and weight are linked by:
z(t) = h - R.?(t)
-1/R . d
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