To find the speed of the ball (m2) when it has fallen a distance h, we can use the principles of energy conservation.
1. First, let's calculate the potential energy at height h for the ball:
Potential energy (PE) = mass (m2) * acceleration due to gravity (g) * height (h)
PE = m2 * g * h
2. Next, let's calculate the work done against friction when the block (m1) slides down the inclined plane:
Work done against friction (W) = force of friction (F) * distance (d)
The distance (d) can be calculated using trigonometry:
d = h / sin(theta)
3. The force of friction (F) is equal to the coefficient of friction (μk) multiplied by the normal force (N).
The normal force (N) can be calculated by resolving the weight of the object perpendicular to the incline:
N = m1 * g * cos(theta)
4. The force of friction (F) can be calculated using:
F = μk * N
5. The work done against friction (W) can be calculated using:
W = F * d
6. The work done against friction (W) is equal to the kinetic energy gained by the ball (m2):
W = 0.5 * m2 * v^2, where v is the velocity of the ball
Now we can solve the equations:
Step 1: Calculate potential energy:
PE = m2 * g * h
Step 2: Calculate the distance (d):
d = h / sin(theta)
Step 3: Calculate the normal force (N):
N = m1 * g * cos(theta)
Step 4: Calculate the force of friction (F):
F = μk * N
Step 5: Calculate the work done against friction (W):
W = F * d
Step 6: Equate work done against friction (W) to the kinetic energy gained by the ball (m2):
0.5 * m2 * v^2 = W
Step 7: Solve for the velocity (v):
v = sqrt(2 * W / m2)
Plug in the given values for m1, m2, μk, g, h, and solve for v.