A 0.40-kg block slides along a small track with elevated ends and a flat central part. The flat part has a length L = 1.84 m. The curved portions of the track are frictionless, but for the flat part the coefficient of kinetic friction is 0.132. The block is released from rest from a height h = 51 cm on the left curved portion of the track. Calculate the maximum height reached by the block on the right curved portion of the track.
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To find the maximum height reached by the block on the right curved portion of the track, we can use the principles of conservation of energy.
Step 1: Calculate the initial potential energy of the block on the left curved portion of the track.
Potential energy (PE) is given by the equation PE = mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height.
m = 0.40 kg
g = 9.8 m/s^2
h = 51 cm = 0.51 m
PE = 0.40 kg * 9.8 m/s^2 * 0.51 m
PE = 1.9992 J (rounded to four decimal places)
Step 2: Calculate the work done by friction on the flat part of the track.
The work done by friction is given by the equation W = μk * N * d, where μk is the coefficient of kinetic friction, N is the normal force, and d is the distance.
μk = 0.132 (coefficient of kinetic friction)
N = mg (normal force), where g is the acceleration due to gravity
d = L (length of the flat part of the track)
N = mg = 0.40 kg * 9.8 m/s^2 = 3.92 N
W = 0.132 * 3.92 N * 1.84 m
W = 0.9583 J (rounded to four decimal places)
Step 3: Calculate the final kinetic energy of the block on the right curved portion of the track.
As the block reaches the right curved portion of the track, it will have lost some of its initial potential energy due to friction. The remaining energy will be converted into kinetic energy.
Kinetic energy (KE) is given by the equation KE = 1/2 * m * v^2, where m is the mass of the block and v is the velocity.
Since the block starts from rest, its initial velocity on the right curved portion of the track will be zero.
KE = 1/2 * 0.40 kg * 0^2
KE = 0 J
Step 4: Calculate the maximum height reached by the block on the right curved portion of the track.
Using the principle of conservation of energy, the total initial energy (PE) is equal to the total final energy (KE) + the work done by friction (W).
PE = KE + W
1.9992 J = 0 J + 0.9583 J + PE_max
Simplifying the equation, we get:
PE_max = 1.9992 J - 0.9583 J
PE_max = 1.0409 J (rounded to four decimal places)
Finally, we can calculate the maximum height using the equation for potential energy:
PE_max = m * g * h_max
h_max = PE_max / (m * g)
h_max = 1.0409 J / (0.40 kg * 9.8 m/s^2)
h_max = 2.66 m (rounded to two decimal places)
Therefore, the maximum height reached by the block on the right curved portion of the track is approximately 2.66 meters.