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A student ties a 10 kg mass onto a fishing line with a breaking strength of 5 newtons, and then starts swinging the mass around over his head. The studenttries very hard to cause the path to be circular. The distance from the center of the circle to the mass is 3 meters.
As time goes on, the student swings the mass faster and faster until the fishing line breaks. What is the tangential velocity of the mass when thefishing line breaks.
Solution:
Centripetal force = mass * (centripetal acceleration), or
f = m * Ar, or
f = m * v^2/r, or
v^2 = f*r/m, or
v = (f*r/m)^(1/2), or
v = (5 newtons * 3 meters/10 kg)^(1/2), or
v = 1.22 m/s
The magnitude of the tangential velocity of the mass when the fishing line breaks is 1.22 meters/second
Check the solution using angular velocity, w
w = v / r, or
w = (1.22 m/s) / 3m = 0.407 radians/second
The angular velocity for a tangential velocity of 1.22 m/s is 0.407 radians/second
Ar = (w^2)*r, or
Ar = (( 0.407 /second)^2)*3m, or
Ar = 0.5 m/s^2
The magnitude of the radial acceleration is 0.5 m/s^2
f = m * Ar, or
f = 10kg*(0.5 m/s^2), or
f = 5 newtons, or
the force equals 5 newtons, which matches the breaking strength of the fishing line.
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