You drag a trunk of mass m across a level floor using a massless rope that makes

Question

You drag a trunk of mass m across a level floor using a massless rope that makes an angle with the horizontal (figure below). Given a kinetic-friction coefficient (?). Find the minimum force needed to move the trunk with constant speed.

Note: The problem does not want you to answer in terms of theta. What it is asking, is if you were allowed to vary the angle at which you pull the trunk, what is the minimum force required.

You will get an expression for the force with theta in it. You then need to minimize this force by varying theta.

Explanation / Answer

From Newton's 2nd law, write equations for the horizontal and vertical forces acting on the trunk. The trunk has mass m and the tension in the rope is T. Since the trunk is pulled at constant speed, the net forces horizontally are zero. They are :

F = 0 = Tcos - µn
Tcos = µn-------->equation (1)

Here, n is the normal force. You will need an expression for the normal force for substitution into equation (1), it may be obtained from the net vertical forces :

F = 0 = n + Tsin - mg

Solved for n :

n = mg - Tsin------->equation (2)

Plugging (2) into (1) :

Tcos = µ(mg - Tsin)

Solving for T:

T = µmg / (cos + µsin)

T is equal to the force required to pull the trunk at constant speed.

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