After placing a 0.500 m dent in the landlords van, the two management students are told they and
their 200 daN beer freezer have to go. The new ramp is inclined at an angle of 32.1 o relative to
the horizontal. The coefficient of static friction between the freezer and the ramp is 0.800. Find the
minimum force required to move the box down the ramp if the force is applied:
(a) parallel to the ramp.
(b) horizontally.
To find the minimum force required to move the box down the ramp, we need to consider the forces acting on the box. There are two cases that need to be considered: the force applied parallel to the ramp and the force applied horizontally.
(a) When the force is applied parallel to the ramp:
In this case, the force of friction acting on the box opposes the applied force. We need to determine the maximum frictional force that can prevent the box from sliding down the ramp. The formula to calculate the maximum frictional force is given by:
f_max = µ_s * N
Where:
- f_max is the maximum frictional force
- µ_s is the coefficient of static friction
- N is the normal force acting on the box
The normal force can be calculated using the formula:
N = m * g * cos(θ)
Where:
- m is the mass of the box
- g is the acceleration due to gravity (approximated as 9.8 m/s²)
- θ is the angle of the ramp
Once we have the maximum frictional force, we can equate it to the applied force to find the minimum force required to move the box:
f_max = F_applied
(b) When the force is applied horizontally:
In this case, the applied force needs to overcome both the force of friction and the component of the gravitational force acting parallel to the ramp. The formula to calculate the minimum required force is:
F_min = µ_k * N + m * g * sin(θ)
Where:
- F_min is the minimum required force
- µ_k is the coefficient of kinetic friction (assumed to be equal to the coefficient of static friction)
- N is the normal force acting on the box
- m is the mass of the box
- g is the acceleration due to gravity (approximated as 9.8 m/s²)
- θ is the angle of the ramp
We can calculate the normal force using the same formula mentioned earlier (N = m * g * cos(θ)) and substitute it into the equation to find the minimum force required.
Remember to convert the angle from degrees to radians before using the trigonometric functions (radians = degrees * π/180).
Using the given values in the problem, substitute them into the above formulas to find the minimum forces required for both cases.