Question

A 0.800-kg ball is tied to the end of a string 1.60 m long and swung in a vertical circle. (a)During one complete circle, starting anywhere, calculate the total work done on the ballby (i) the tension in the string and (ii) gravity. (b) Repeat part (a) for motion along thesemicircle from the lowest to the highest point on the path.

Answer 1

Answer: a) i) 0 ii) 0  b) i) 0 ii) -25.1 J

Explanation:

a) For the first part, as work is defined as the process through an applied force produces a displacement in the direction of the movement, if the start and the end points are the same (as for a complete circle), this means that there is no displacement, i.e. total work (for any applied force) is 0.

b) Taking only a semicircle from the lowest to the highest point on the path, as tension force is always perpendicular to the displacement, we can conclude that the tension force does no work.

Now, for gravity, the displacement, is just the difference in height between the highest and lowest point, which is equal to twice the length of the string, and is a vector with only a vertical component, as follows:

Δy = y₂ - y₁ (by definition of displacement)

Now, the work done by gravity is just the product of the weight of  the ball, times the vertical displacement (as the work done by gravity is independent of the trajectory) , as follows :

W = m. g. (y₂-y₁) = 0.8 Kg. 9.8 m/s². (-2. 1.6 m) = -25.1 J

(The minus sign is due to gravity and the displacement point in opposite directions, in this case we take as positive the downward direction).