A mass of 14 kg is moving at a speed of 39 m/s at the bottom of incline which is inclined at an angle of 51.6 degrees. The incline has friction and the coefficient of kinetic friction is 0.6. The mass travels up the incline and stops due to both gravity and friction and then slides back down the incline. When the object slides back down the incline, how fast is it moving at the bottom in m/s?
To determine the speed at which the mass is moving at the bottom of the incline when sliding back down, we need to consider the forces acting on the object and the work done.
First, let's analyze the forces:
1. Gravitational force (mg): This force acts downward and can be calculated by multiplying the mass (m) by the acceleration due to gravity (g ≈ 9.8 m/s^2).
F_gravity = mg
2. Normal force (N): This force acts perpendicular to the incline and balances the component of the gravitational force.
F_normal = mg*cos(θ)
3. Frictional force (F_friction): This force opposes the motion and acts parallel to the incline. The frictional force can be calculated by multiplying the coefficient of kinetic friction (μ) by the normal force.
F_friction = μ*N
4. Net force or horizontal force (F_net): This force is the difference between the gravitational force and the frictional force.
F_net = mg*sin(θ) - μ*N
Next, we need to calculate the work done by the net force. The work done is equal to the change in kinetic energy of the mass.
Work = ΔKE
Finally, we can use the work-energy theorem to find the speed of the object at the bottom of the incline. The work done is equal to the change in kinetic energy.
Work = ΔKE = KE_final - KE_initial
Since the object starts from rest at the top of the incline and returns to the bottom with the same speed, the initial kinetic energy is zero. Therefore, the work done is equal to the final kinetic energy.
Work = KE_final
Now, let's solve the problem step by step:
1. Calculate the gravitational force:
F_gravity = 14 kg * 9.8 m/s^2 = 137.2 N
2. Calculate the normal force:
N = mg*cos(θ) = 14 kg * 9.8 m/s^2 * cos(51.6°) = 95.27 N
3. Calculate the frictional force:
F_friction = μ*N = 0.6 * 95.27 N = 57.16 N
4. Calculate the net force:
F_net = mg*sin(θ) - F_friction = 14 kg * 9.8 m/s^2 * sin(51.6°) - 57.16 N = 44.42 N
5. Calculate the work done:
Work = F_net * distance
To find the distance traveled up the incline and back down, we need to consider the vertical displacement. The vertical displacement is given by:
distance = 2 * height = 2 * (height*sin(θ))
Plug in the values:
distance = 2 * (39 m/s * sin(51.6°)) = 49.58 m
Therefore,
Work = 44.42 N * 49.58 m = 2201.25 J
6. Calculate the final velocity (v_final) using the work-energy theorem:
Work = KE_final
2201.25 J = (1/2) * 14 kg * v_final^2
Solving for v_final gives us:
v_final = sqrt((2 * Work) / (m))
v_final = sqrt((2 * 2201.25 J) / (14 kg)) ≈ 11.62 m/s
Hence, the speed at which the object is moving at the bottom of the incline when sliding back down is approximately 11.62 m/s.