Describe constant-velocity and static-equilibrium conditions by requiring zero net force and balancing force components in blocks, ropes, pulleys, and extended systems.
Blocks of mass m and 2 m are connected by a light string and placed on a frictionless inclined plane that makes an angle θ 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 T2 exerted on the block of mass m by the string. Express your answers in terms of m,θ, and physical constants, as appropriate.
The mass M for which the system can remain in equilibrium. Express your answers in terms of m,θ, and physical constants, as appropriate.
For using an attempt at a correct statement of Newton's second law for the two blocks
1 point
F_net =(m+2 m) a=0 T_2-(m+2 m) g =0
For a correct answer with supporting work
1 point
T_2=3 m g
LO INT-1.D, SP 5.E 1 point
The mass M for which the system can remain in equilibrium. Express your answers in terms of m,, and physical constants, as appropriate.
For using a correct statement of Newton's second law for the whole system to derive an
expression for M1 point
F_net =m_tot a M g-2 m g -m g =0M=3 m
Alternate Solution Alternate Points
For a correct statement of Newton's second law for the sphere and an answer consistent
with part (b)(i)
1 point
T_2-M g=0 M g=T_2 M=T_2 / gM=3 m