Question

A light string of total length L is wrapped completely around the axle of a spinning toy top. The radius of the axle is r and the rotational inertia of the top is I. The top is initially at rest and the axle is held vertically with respect to a horizontal table. At time t = 0 a force with constant magnitude F pulls the string outwards from the axle of the top, which causes the top to rotate. After a time T the string loses contact with the axle of the top and the top spins on the table with an angular speed ω (omega). Assume the top always remains upright and there is negligible friction between the top and the table. The original string is replaced with a new string of length 2L. The string is again pulled with a constant force of magnitude F. When the string loses contact with the top, the angular speed of the top is ω A . Is ω A greater than, less than, or equal to ω ? Justify your answer without citing or manipulating equations. The original string of length L is used to spin a second top that also has an axle with radius r, but a larger rotational inertia I B = 2 I . The string is pulled with the same constant force of magnitude F. When the string loses contact with the top, the angular speed of the second top is ω

Answer 1

Let's call the first angular speed , the second one and the third one

The angular acceleration in the first case is

Therefore,the angular velocity is

In the second case, the angular acceleration is the same, except that time it acts for is double,as double the length of rope is present.

We can see that

In the third case, the string is the same but top is replaced.

The new angular acceleration is

The angular velocity will therefore be