Block a of mass m is on a horizontal surface of negligible friction. An identical block b is attached to block a by a light string that passes over an ideal pulley. The tension force exerted on block a after the system is released from rest has magnitude t. Block b is then replaced by a block of mass 2m and the system again released from rest. Which of the following is a correct expression for the tension force now exerted on block a?.

When block B has mass m,

• the net force on block A is

∑ F = -T = ma

• the net force on block B is

∑ F = T - mg = ma

(note that we take the direction in which block A is pulled and in which block B is pulled downward to be negative)

By adding these two equations, we eliminate the tension T and find

-T + (T - mg) = ma + ma   ⇒   -mg = 2ma   ⇒   a = -g/2

Then the tension has an initial magnitude of

T = -ma = mg/2

Now replace block B with a block of mass 2m. Then

• the net force on block A is

∑ F = -T' = ma

• the net force on block B is

∑ F = T' - 2mg = 2ma

By eliminating T', we find

-T' + (T' - 2mg) = ma + 2ma   ⇒   -2mg = 3ma   ⇒   a = -2g/3

so that

T' = -m (-2g/3) = 2mg/3

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