A mass of 0.475 kg, on a frictionless table, is attached to a string which passes through a hole in the table. The hole is at the center of a horizontal circle in which the mass moves with constant speed. The radius of the circle is 0.510 m and the speed of the mass is 13.0 m/s. It is found that drawing the string down through the hole and reducing the radius of the circle to 0.370 m has the effect of multiplying the original tension in the string by 4.63. Compute the total work (in joules) done by the string on the revolving mass during the reduction of the radius.
To compute the total work done by the string on the revolving mass during the reduction of the radius, we need to find the change in potential energy of the mass as it moves from the initial circle with a radius of 0.510 m to the final circle with a radius of 0.370 m.
The potential energy of an object in a circular motion is given by the formula:
U = (1/2) * m * v^2
where U is the potential energy, m is the mass of the object, and v is the velocity of the object.
In this case, the speed of the mass remains constant at 13.0 m/s. Therefore, the change in potential energy can be calculated using the formula:
ΔU = (1/2) * m * (v_final^2 - v_initial^2)
To find the change in tension in the string, we can use the fact that reducing the radius has the effect of multiplying the original tension by 4.63. Let's assume the original tension in the string is T.
T_final = 4.63 * T
As the mass moves from the initial circle to the final circle, the tension in the string does work on the mass. This work is equal to the change in potential energy:
Work = ΔU = (1/2) * m * (v_final^2 - v_initial^2)
Substituting the given values:
mass (m) = 0.475 kg
velocity initial (v_initial) = 13.0 m/s
velocity final (v_final) = 13.0 m/s
radius initial (r_initial) = 0.510 m
radius final (r_final) = 0.370 m
We can plug the values into the formula:
Work = (1/2) * 0.475 kg * ((13.0 m/s)^2 - (13.0 m/s)^2)
Simplifying this equation, we find:
Work = 0.0 J
So, the total work done by the string on the revolving mass during the reduction of the radius is 0.0 joules.