Q.318. Mechanics - Friction

Question 318.
A 29.5 kg block is connected to an empty 1.00 kg bucket by a cord running over a frictionless pulley. The coefficient of static friction between the table and the block is 0.485 and the coefficient of kinetic friction between the table and the block is 0.320. Sand is gradually added to the bucket until the system just begins to move.
(a) Calculate the mass of sand added to the bucket.
(b) Calculate the acceleration of the system. (downward).

Answer 318.
Maximum static frictional force on the block
= μ(s) * mg
= (0.485) * (29.5) * (9.81) N
≈ 140.4 N.

(a)
Let m = mass of sand to be added in kg to start the motion of the block
=> (m + 1) * 9.81 = 140.4
=> m ≈ 13.31 kg

(b)
If T = tension in the cord once the motion starts, then
acceleration of the bucket = acceleration of the block = a
=> (13.31 + 1) * 9.81 - T = (13.31 + 1) a ... (1)
and T - (29.5) * (9.81) * (0.320) = 29.5 a ... (2)
Adding (1)and (2),
(13.31 + 1) * 9.81 - (29.5) * (9.81) * (0.320) = (13.31 + 1) a + 29.5a
=> 43.81 a = 47.77
=> a = 1.09 m/s^2.

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