A 10 kg box is being pushed up a slippery ramp as shown in Figure
2.12. The coefficient of friction between the box and the ramp
is just μ = 0.1.
(a) What force does the man need to apply to the box to keep it
traveling up the ramp at a steady speed?
(b) What fraction of the weight force of the box is this?
To find the force that the man needs to apply to the box to keep it traveling up the ramp at a steady speed, we need to consider the forces acting on the box.
(a) The forces acting on the box are:
1. The weight force (mg), which acts vertically downward.
2. The normal force (N), which acts perpendicular to the surface of the ramp.
3. The force of friction (f), which acts parallel to the surface of the ramp and opposes the motion of the box.
Since the box is traveling at a steady speed, we know that the net force acting on the box is zero. This means that the force of friction must be equal in magnitude and opposite in direction to the component of the weight force parallel to the ramp.
The component of the weight force parallel to the ramp can be calculated using trigonometry. In this case, the angle of the ramp is not given, so let's assume it is θ.
The component of the weight force parallel to the ramp (F_parallel) can be calculated using the equation:
F_parallel = mg * sin(θ)
The force of friction (f) can be calculated using the equation:
f = μ * N
Since the box is not accelerating, we can write the equation:
F_parallel - f = 0
Substituting the expressions for F_parallel and f, we get:
mg * sin(θ) - μ * N = 0
We know that the normal force (N) is equal in magnitude and opposite in direction to the component of the weight force perpendicular to the ramp. So, N = mg * cos(θ).
Substituting this expression for N in the equation, we get:
mg * sin(θ) - μ * mg * cos(θ) = 0
Now, solving for the force applied by the man (F_applied):
F_applied = μ * mg * cos(θ)
(b) To find the fraction of the weight force, we can divide the force applied by the man by the weight of the box (mg) and express it as a percentage.
Fraction of weight force = (F_applied / mg) * 100
Therefore, to calculate the force the man needs to apply to the box (F_applied) and the fraction of the weight force, we need to know the angle of the ramp (θ).