A 90 kg box is pushed by a horizontal force F at constant speed up a ramp inclined at 28°, as shown. Determine the magnitude of the applied force.
when the ramp is frictionless.
when the coefficient of kinetic friction is 0.18.
Downward component of weight (along incline)
= mg(sin(θ))
Normal component of weight
= mg(cos(&theta));
Upward component of horizontal force (along incline)
= F(cos(θ))
Normal component of horizintal force
= F(sin(θ))
Coefficient of kinetic friction = μ
Total normal reaction
N= mg(cos(θ)+F(sin(θ))
Frictional resistance
=μ(N)
For equilibrium along the inclined plane
upward force = downward force
mg(sin(θ))+μN=F(cos(θ))
or
mg(sin(θ))+μ(mg(cos(θ)+F(sin(θ)))=F(cos(θ))
Substituting
m=90 kg
g=9.8 m/s²
θ=28°
μ=0.18
Solve for F
With μ=0
solve for F.
I get approximately 7*10^2N and 4.7*10^2N for the two cases.
To determine the magnitude of the applied force in each scenario, we can use Newton's second law of motion and analyze the forces acting on the box.
When the ramp is frictionless:
In this case, the only forces acting on the box are its weight (mg) and the applied force (F) parallel to the ramp. These forces can be resolved into components perpendicular and parallel to the ramp.
The weight can be resolved into two components:
- The component perpendicular to the ramp is mg * cos(28°).
- The component parallel to the ramp is mg * sin(28°).
The applied force can also be resolved into two components:
- The component perpendicular to the ramp is F * sin(28°).
- The component parallel to the ramp is F * cos(28°).
Since the box is moving at constant speed, the net force acting on it in the direction parallel to the ramp is zero. Therefore, the magnitude of the applied force (F) can be found by equating the component of the weight parallel to the ramp to the component of the applied force parallel to the ramp:
F * cos(28°) = mg * sin(28°)
Substituting the given values:
F * cos(28°) = (90 kg) * (9.8 m/s^2) * sin(28°)
Simplifying the equation:
F = [(90 kg) * (9.8 m/s^2) * sin(28°)] / cos(28°)
To find the value of F, you can calculate this expression using a scientific calculator or software.
When the coefficient of kinetic friction is 0.18:
In this case, the force of friction (f_k) opposes the motion of the box, and we need to consider it in our analysis.
The force of friction can be calculated using the equation:
f_k = μ_k * N
Where:
μ_k is the coefficient of kinetic friction,
N is the normal force perpendicular to the ramp.
The normal force (N) can be determined by considering the component of the weight perpendicular to the ramp:
N = mg * cos(28°)
Substituting the given values:
N = (90 kg) * (9.8 m/s^2) * cos(28°)
The magnitude of the applied force is equal to the sum of the force in the direction parallel to the ramp and the force of friction, since the box is moving at a constant speed:
F = F_parallel + f_k
F_parallel = mg * sin(28°)
Substituting the given values and calculating:
F_parallel = (90 kg) * (9.8 m/s^2) * sin(28°)
f_k = (0.18) * N
Substituting the value of N and calculating:
f_k = (0.18) * [(90 kg) * (9.8 m/s^2) * cos(28°)]
Finally, the total applied force F is:
F = F_parallel + f_k
You can calculate the value of F by adding F_parallel and f_k using a calculator or software.