As shown in the figure below, a 2.25-kg block is released from rest on a ramp of height h. When the block is released, it slides without friction to the bottom of the ramp, and then continues across a surface that is frictionless except for a rough patch of width 15.0 cm that has a coefficient of kinetic friction μk = 0.490. Find h such that the block's speed after crossing the rough patch is 3.40 m/s.
To find the height `h` such that the block's speed after crossing the rough patch is 3.40 m/s, we can use the principles of conservation of energy.
Let's break down the problem step by step:
Step 1: Calculate the potential energy loss on the incline:
The block loses potential energy as it slides down the ramp. The potential energy lost is given by the formula:
ΔPE = m * g * h
where:
m = mass of the block = 2.25 kg
g = acceleration due to gravity = 9.81 m/s^2 (approximate)
So, ΔPE = 2.25 kg * 9.81 m/s^2 * h
Step 2: Calculate the work done by friction on the rough patch:
The work done by friction on the block as it crosses the rough patch results in the loss of kinetic energy. The work done by friction is given by:
Work = force * distance * coefficient of kinetic friction
The force can be calculated using the equation: force = mass * acceleration
Since the block is moving with a constant speed across the rough patch, the force of friction is equal to the force applied to overcome it (static friction is not applicable here). Thus, force = mass * acceleration = m * g.
Therefore, Work = m * g * distance * coefficient of kinetic friction
The distance is given in centimeters, so we need to convert it to meters: distance = 15.0 cm * 0.01 m/cm
Step 3: Calculate the change in kinetic energy:
The change in kinetic energy is equal to the initial kinetic energy minus the work done by friction.
Initial kinetic energy = 0 (since the block is released from rest)
Work = m * g * distance * coefficient of kinetic friction
Therefore,
ΔKE = 0 - m * g * distance * coefficient of kinetic friction
Step 4: Equate the potential energy loss on the incline to the change in kinetic energy and solve for `h`:
To find the height `h`, which is the difference in height between the starting point and the end point, we can equate the potential energy loss on the incline to the change in kinetic energy:
ΔPE = ΔKE
2.25 kg * 9.81 m/s^2 * h = -2.25 kg * 9.81 m/s^2 * distance * coefficient of kinetic friction
Substituting the values, we get:
2.25 kg * 9.81 m/s^2 * h = -2.25 kg * 9.81 m/s^2 * (15.0 cm * 0.01 m/cm) * 0.490
Now, we can solve this equation for `h`.