At a winter fair a 74.9-kg stunt man is shot from a horizontal cannon that rests

Question

At a winter fair a 74.9-kg stunt man is shot from a horizontal cannon that rests at the edge of a frozen lake. The human projectile is cast onto the smooth ice, slides some distance, and grabs the end of a long rope whose other end is attached to a pivot that is firmly anchored to the ice. The rope initially lies perpendicular to the stunt man's line of motion, and when he grabs it, he starts sliding in circular motion about the pivot and continues to revolve until he comes to rest. A tonometer on the rope indicates a tension of 851 N at the beginning of the circular motion. With a coefficient of kinetic friction of 0.0421, how many revolutions does the stunt man make? Take g = 9.81 m/s^2. Assume the rope supplies all the centripetal force.

Explanation / Answer

Number of revolutions = Total distance ÷ Circumference
Circumference = 2 * * r
Number of revolutions = Total distance ÷ 2 * * r

The tension force is the force which causes the stunt man to move in circular path. So, the tension force is the centripetal force. The friction force causes the stunt man to decelerate as he moves around the circle. This means the velocity of the stunt man is continually decreasing. Since the centripetal force is dependent on the velocity, and the tension force is equal to the centripetal force, the tension force is continually decreasing. So, the stunt man will stop moving, when the tension force is 0 N.


Tension = m * vi^2/r
851 = 74.9 * vi^2/r
851 ÷ 74.9 = vi^2/r
This is the initial centripetal acceleration.

The friction force is perpendicular to the radius of the circle. So, the friction force is perpendicular to the tension force.
So the friction force is producing the torque which causes the angular velocity of the stunt man to decrease.
Friction force = 0.0421 * 74.9 * 9.81

Friction force = mass * deceleration
Friction force = 0.0421 * 74.9 * 9.81
0.0421 * 74.9 * 9.81 = 74.9 * deceleration
Deceleration = 0.0421 * 9.81 = 0.413001 m/s^2
As the stunt man moves around the circle, his velocity decreases at the rate of 0.413001 m/s each second.
Since we don’t know the time, use the following equation.
vf^2 = vi^2 + 2 * a * d
0 = vi^2 + 2 * -0.413001 * d
vi^2 = 0.826002 * d

851 ÷ 74.9 = vi^2/r
vi^2 = r * 851 ÷ 74.9

Set the two equations for vi^2 equal to each other.
0.826002 * d = r * 851 ÷ 74.9
Divide both sides by0.826002
d = r * (851 ÷ 74.9) ÷ 0.826002
This is the total distance.

Number of revolutions = Total distance ÷ 2 * * r
Number of revolutions = [r * (851 ÷ 74.9) ÷ 0.826002] ÷ 2 * * r
Number of revolutions = [(851 ÷ 74.9) ÷ 0.826002] ÷ 2*
The number of revolutions is approximately 1.67.

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