A large box of mass M is pulled across a horizontal, frictionless surface by a h

Question

A large box of mass M is pulled across a horizontal, frictionless surface by a horizontal rope with tension T. A small box of mass m sits on top of the large box. The coefficients of static and kinetic friction between the two boxes are and mu k, respectively. Part A Find an expression for the maximum tension Tmax for which the small box rides on top of the large box without slipping. Express your answer in terms of the variables M, m, mu s, and appropriate constants. Part B A horizontal rope pulls a 12 kg wood sled across frictionless snow. A 6.0 kg wood box rides on the sled. What is the largest tension force for which the box doesn't slip? Assume that mu s = 0.50. Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

a) Assuming the small box doesn't slip, the tension T accelerates mass M+m according to

T = (M+m)*a

For box m to ride along and not slip, the force of static friction must be such that m accelerates at the value a.

Ff_s = m*a

The force of static friction is given by

Ff_s = mu_s*N = mu_s*m*g

We can write an expression for the maximum acceleration

Ff_s = m*a = mu_s*m*g

a = mu_s*g

Using that in the first expression, the max T that avoids slippage is

T = (M+m)*mu_s*g

mu_k is not required.

b) This is equivalent to the above situation.

T = (12 kg+6 kg)*mu_s*9.8 m/s^2

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