A 0.30-kg block slides along a small track with elevated ends and a flat central part. The flat part has a length L = 1.46 m. The curved portions of the track are frictionless, but for the flat part the coefficient of kinetic friction is 0.101. The block is released from rest from a height h = 95 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.
To solve this problem, we will use the law of conservation of mechanical energy. At any point along the track, the total mechanical energy of the block is given by the sum of its gravitational potential energy and its kinetic energy.
1. Calculate the initial potential energy of the block on the left curved portion of the track:
- The mass of the block is given as 0.30 kg.
- The height of the block on the left curved portion of the track is given as h = 95 cm = 0.95 m.
- The acceleration due to gravity is 9.8 m/s^2.
- The initial potential energy of the block is given by: P.E. = mgh = (0.30 kg)(9.8 m/s^2)(0.95 m).
2. Calculate the initial kinetic energy of the block on the left curved portion of the track:
- The block is released from rest, so its initial velocity is 0 m/s.
- The initial kinetic energy of the block is given by: K.E. = 0.5mv^2 = 0.5(0.30 kg)(0 m/s)^2.
3. Calculate the initial total mechanical energy of the block on the left curved portion of the track:
- The initial total mechanical energy is given by: E_i = P.E. + K.E.
4. Calculate the final kinetic energy of the block on the right curved portion of the track:
- We know that the block reaches a maximum height on the right curved portion of the track. At this point, all of its initial kinetic energy is converted into gravitational potential energy.
- The mass of the block is 0.30 kg.
- The acceleration due to gravity is 9.8 m/s^2.
- The final velocity of the block at the maximum height is 0 m/s.
- The final kinetic energy of the block is 0 J.
5. Calculate the final potential energy of the block on the right curved portion of the track:
- The mass of the block is 0.30 kg.
- The height of the block on the right curved portion of the track is unknown, denoted as h_r.
- The acceleration due to gravity is 9.8 m/s^2.
- The final potential energy of the block is given by: P.E. = mgh_r.
6. Calculate the final total mechanical energy of the block on the right curved portion of the track:
- The final total mechanical energy is given by: E_f = P.E. + K.E.
7. Apply the law of conservation of mechanical energy:
- According to the conservation of mechanical energy, the initial and final total mechanical energies must be equal: E_i = E_f.
- Set the equations for the initial and final total mechanical energies equal to each other and solve for h_r.
Plugging in the values and solving the equation, you will find the maximum height reached by the block on the right curved portion of the track.