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 mu and mu_k respectively. Find an expression for the maximum tension T_max 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 and appropriate constants. T (M + m) mu_s g A horizontal rope pulls a 9.0 kg wood sled across frictionless snow. A 5.6 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 = (M+m)*mu_s*g
T = (9 kg+5.6 kg)*0.50*9.8 m/s^2
T=71.54 N

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