A stuntman whose mass is 77 kg swings from the end of a 5.0-m-long rope along the arc of a vertical circle. Assuming that he starts from rest when the rope is horizontal, find the magnitudes of the tensions in the rope that are required to make him follow his circular path at each of the following points. (a) at the beginning of his motion kN (b) at a height of 1.5 m above the bottom of the circular arc kN (c) at the bottom of the arc kN
To find the tensions in the rope at different points along the circular path, we need to consider the forces acting on the stuntman at each point.
(a) At the beginning of his motion when the rope is horizontal:
At this point, the tension in the rope provides the centripetal force needed to keep the stuntman moving in a circular path. The net force acting on the stuntman is the tension in the rope. Since he starts from rest, there is no initial velocity.
Using the centripetal force formula:
Fnet = Tension = m*v^2/r
where m is the mass of the stuntman, v is the velocity of the stuntman, and r is the radius of the circular path.
At the beginning of his motion, the radius of the circular path is equal to the length of the rope, which is 5.0 m. The velocity of the stuntman can be calculated using the equation: v = sqrt(2*g*h) where h is the height from which the stuntman falls.
Since he starts from rest, the height from which he falls is equal to the radius of the circle.
v = sqrt(2*g*5.0) = sqrt(2*9.8*5.0) = 9.9 m/s
Substitute the values into the centripetal force formula:
Tension = m*v^2/r = 77 * (9.9)^2 / 5.0 = 1522.86 N = 1.52 kN
Therefore, at the beginning of his motion, the tension in the rope is 1.52 kN.
(b) At a height of 1.5 m above the bottom of the circular arc:
At this point, the tension in the rope must provide the centripetal force as well as counteract the weight of the stuntman.
The net force acting on the stuntman is the tension in the rope minus the weight of the stuntman:
Tension - weight = m*v^2/r
Weight = m*g = 77*9.8 = 754.6 N
The velocity of the stuntman at this point can be calculated using the conservation of energy, taking the initial height to be 1.5 m:
m*g*h_initial = 1/2*m*v^2
77*9.8*1.5 = 1/2*77*v^2
v = sqrt(2*9.8*1.5) = 6.89 m/s
Substitute the values into the equation:
Tension - weight = m*v^2/r
Tension - 754.6 = 77 * (6.89)^2 / 5.0
Tension = 1510.08 N = 1.51 kN
Therefore, at a height of 1.5 m above the bottom of the circular arc, the tension in the rope is 1.51 kN.
(c) At the bottom of the arc:
At the bottom of the arc, the tension in the rope must also provide the centripetal force and counteract the weight of the stuntman. The velocity of the stuntman at the bottom of the arc can be calculated using the conservation of energy:
m*g*h_final = 1/2*m*v^2
77*9.8*0 = 1/2*77*v^2
v = 0 m/s
Substitute the values into the equation:
Tension - weight = m*v^2/r
Tension = weight + m*v^2/r
Tension = 77*9.8 + 77*0 / 5.0
Tension = 754.6 N
Therefore, at the bottom of the arc, the tension in the rope is equal to the weight of the stuntman, which is 754.6 N or 0.75 kN.