Question

A rope is tied to a tree limb and is used by a swimmer to swing into the water. The person starts from rest with the rope held in the horizontal position, as shown in the figure on the power point file attached, swings downward and then let go of the rope. His initial height is h0 = 415 m and final height is hf = 105 m. If the force due to air resistance is neglected,

(30%) What is the total work done on the person by the tensional force of the rope? You have to explain in details how you arrive at your answer to get the maximum credit. Simply put down the answer without supporting arguments will NOT obtain good credit, even if the answer is correct!

(30%) What is the speed of the swimmer as he let go of the rope? You have to explain in details how you arrive at your answer to get the maximum credit. Simply put down the answer without supporting arguments will NOT obtain good credit, even if the answer is correct!

(40%) If the speed of the swimmer is measured to be 67.8 m/s, and the mass of the swimmer is 75.0 kg, find the total work done on the swimmer by the frictional force – air resistance during this swinging process. You have to explain in details how you arrive at your answer to get the maximum credit. Simply put down the answer without supporting arguments will NOT obtain good credit, even if the answer is correct!

Answer 1

The force of tension (T) acts radially inward toward the center of the circle. The tension is always perpendicular to the circular path of the motion.

The direction of the displacement vector (ds) of the person acts tangentially and makes right angle with the tension. So the component of the displacent in the direction of the tension is 0.

The total work done on the person by the tensional force of the rope

= T . ds = |T| |ds| cos = 0

At the initial position, the height h0 = 415 m

                           and, the speed v0 = 0

At the final position, the height hf = 105 m

                                 the speed vf = ?

Mass of the swimmer (m) = 75 kg

From, the principle of conservation of the mechanical energy, we can write,

   ------------   (1)

where, g = acceleration due to gravity = 9.81 ms-2

From (1) we get,

or, ms-1

The speed of the swimmer is measured to be (vm) = 67.8 ms-1

So it is less than the estimated value vf which was estimated neglecting the frictional force. In the presence of friction, it will oppose the motion of the person. So, some of the kinetic energy will be expended to overcome the frictional forces.

Therefore, the difference between the estimated and observed kinetic energy is same as the total work done on the swimmer by the frictional force,

Therefore, the total work done on the swimmer by the frictional force is 55710 J