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. (a) The string is again wrapped around the axle of the top. The string is pulled with a new constant force FA=2F , and at time TA the string loses contact with the top. Is TA greater than, less than, or equal to T ? Justify your answer without citing or manipulating equations.

(b) The string is wrapped around a second top with smaller rotational inertia IB=12I . The axle of the second top is the same radius as the radius of the axle of the first top. The same constant force with magnitude F is applied to the string, and at time TB the string loses contact with the top. Is TB greater than, less than, or equal to T ? Justify your answer without citing or manipulating equations.

(c) A student derives the following equation to predict the amount of time over which the string is in contact with the top: T=−√(2IL/Fr^2) . Whether or not this equation is correct, does it match your reasoning in parts (a) and (b)? Once the string has been pulled for a certain amount of time T , the string comes off and the top spins on the table. Now consider the case in which friction between the top and the table is not negligible, and the friction exerts a constant torque on the top about a vertical axis through the center of the top.

(d) Describe any changes in the top’s angular momentum about a vertical axis through the center of the top after the string loses contact with the top. Explain your answer.