Physics force question

Question

You have a box of books on the floor with a total mass of 100-kg. You attach a rope to the box at an angle of 30 degrees above the horizontal. If the coefficient of kinetic friction is .22 between the box and the floor, what force is needed to slide the box at a constant velocity?

Answer 1

To move at constant velocity, the net force must be zero. During this event, you would have a gravitational force downward, a normal force upward. friction opposite motion, and therefore opposite the pulling force from the rope. Lets assume the rope is up and to the right. That would mean that friction is to the left. To determine the friction force, you need to know the normal force since:

F_F = μ F_N

In the vertical direction, there is no acceleration, so all the forces must be balanced. We have the normal force up, a component of the tension in the rope up, and gravity down. To find the component of the tension, we need to use trigonometry. Therefore in the vertical direction:

F_Net = F_N + F_T(sin θ) - F_G = 0

We can finde the force of gravity using:

F_G = mg

To create the equation:

F_N + F_T(sin θ) = mg (Eqn 1)

In the horizontal direction, there is also no acceleration, so the forces in the horizontal direction must also be balanced. Again, to find the horizontal component of the tension, we must use trigonometry. So in the horizontal direction:

F_Net = F_T(cos θ) - F_F = 0

We also know that for kinetic friction:

F_F = μ F_N

so we can combine those two equations:

F_T (cosθ) - μ F_N = 0 (Eqn 2)

Since we know the angle θ is 30 degrees, we now have two equations and two unknowns, F_T and F_N. Since we only are asked to find F_T, we can solve equation 1 for F_N:

F_N = mg - F_T(sin θ)

And substitute that into equation #2:

F_T(cosθ) - μ(mg - F_T(sin θ) = 0

Distributing the "-μ" gives us:

F_T(cos θ) - μmg + μF_T(sin θ) = 0

Rearranging this gives:

F_T(cos θ) + μF_T(sin θ) = mg

Factoring for F_T and dividing gives:

F_T = mg / (cos θ + μsin θ)

Plugging in values:

F_T = (100 kg)(10 N/kg) / [(cos 30) + 0.22 (sin 30)]

So:

F_T = 1000 N / [0.866 + 0.22(0.5)] = 225.4 N