AP Physics C Mechanics 2.4: Newton’s First Law

Blocks of mass m and 2 m are connected by a light string and placed on a frictionless inclined plane that makes an angle $\theta$ with the horizontal, as shown in Figure 1 above. Another light string connecting the block of mass m to a hanging sphere of mass M passes over a pulley of negligible mass and negligible friction. The entire system is initially at rest and in equilibrium.

Derive expressions for the magnitude of each of the following. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figures in part (a).

The force $T_{2}$ exerted on the block of mass m by the string. Express your answers in terms of $m, \theta$, and physical constants, as appropriate.

The mass M for which the system can remain in equilibrium. Express your answers in terms of $m, \theta$, and physical constants, as appropriate.