Question

You slide a box of books at constant speed up a 30° ramp, applying a force of 200 n directed up the slope. the coefficient of sliding friction is 0.18. (a) how much work have you done when the box has risen 1 m vertically? (b) what's the mass of the box?

Answer 1

The box is moving at constant speed, so acceleration=0 and the equation of the forces acting on the box along the ramp becomes:

$F-F_f-W_p = 0$ (1)

where F=200 N is the force that pushes the box, Ff is the frictional force, and Wp is the component of the weight parallel to the ramp.

The frictional force can be written as

$F_f = \mu m g cos \theta$

where $\mu=0.18$ is the coefficient of friction, m is the mass of the box, $g=9.8 m/s^2$ is the frictional force and $\theta=30^{\circ}$ is the angle of the ramp.

The component of the weight along the ramp is

$W_p = mg sin \theta$

Substituting this into eq.(1), we have

$F-\mu mg cos \theta- mg sin \theta=0$

and we can find the mass of the box:

$m=\frac{F}{\mu g cos \theta + g sin \theta}=\frac{200 N}{(0.18)(9.8 m/s^2)(cos 30^{\circ})+(9.8 m/s^2)(sin 30^{\circ})}=31.1 kg$

Now we can also find the work done on the box. this is given by the gain in potential energy of the box, since there is no change in kinetic energy (the speed of the box is constant). Since the box has risen vertically by 1 m, the gain in potential energy (and the work done) is

$W=\Delta U=mg \Delta h=(31.1 kg)(9.8 m/s^2)(1 m)=304.8 J$

So, the two answers are

(a) 304.8 J

(b) 31.1 kg

Answer 2

A. Work = integral of force across a path. Since the force is constant, just multiply them: force*distance = 200N * (1m*sin(30)) = 100 Nm = 100 J B. Use Newton: F=ma. For this, we need to use the sum of all the forces, not just the work we apply on the books, but gravity and friction too. Constant velocity means zero acceleration, which means the sum of forces is zero. Draw a free body diagram with the three forces and you'll see that the uphill force = the downhil force => 200N = NormalForce*Coefficient + Gravity*mass*sin(30) 200N = Gravity*mass*cos(30)*coef. + Gravity*mass*sin(30) 200N = 9.81*mass*(cos(30)*0.18 + sin(30)) mass = 200/(9.81*(cos(30)*0.18 + sin(30))) m = 31.23 kg