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 ?s and ?k, respectively.

A- Find an expression for the maximum tension Tmax for which the small box rides on top of the large box without slipping.

B- A horizontal rope pulls a 11kg wood sled across frictionless snow. A 4.4kg wood box rides on the sled. What is the largest tension force for which the box doesn't slip? Assume that ?s=0.50.

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 = (11 kg+4.4 kg)*mu_s*9.8 m/s^2

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