A 0.50-kg block slides along a small track with elevated ends and a flat central part. The flat part has a length L = 1.85 m. The curved portions of the track are frictionless, but for the flat part the coefficient of kinetic friction is 0.103. The block is released from rest from a height h = 96 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.
PE=KE = W(fr)+KE1
m•g•H = k•m•g•L +KE1
KE1= m•g•H – k•m•g•L
KE1= PE1
PE1 = m•g•h
m•g•H – k•m•g•L = m•g•h
h = H - k•L=0.96 – 0.103•1.85=0.769 m
Well, let's see if I can track down an answer for you! Since the block is released from rest, we can use the principle of conservation of mechanical energy. At the top of the curved section, all of the gravitational potential energy is converted into kinetic energy.
The initial potential energy can be calculated using the equation PE = mgh, where m is the mass (0.50 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (0.96 m). So, the initial potential energy is PE = (0.50 kg)(9.8 m/s^2)(0.96 m) = 4.75 J.
Since there is no friction on the curved sections, all of the initial potential energy is converted into kinetic energy at the bottom of the curve. So, the kinetic energy at the bottom of the curved section is KE = 4.75 J.
Now, as the block slides on the flat part, there is friction acting against its motion. The work done by friction can be calculated using the equation W = μk * N * d, where μk is the coefficient of kinetic friction (0.103), N is the normal force (equal to the weight of the block), and d is the distance (1.85 m).
The normal force is equal to the weight of the block, which can be calculated using the equation N = mg, where m is the mass (0.50 kg) and g is the acceleration due to gravity (9.8 m/s^2). So, N = (0.50 kg)(9.8 m/s^2) = 4.9 N.
The work done by friction can be calculated as W = (0.103)(4.9 N)(1.85 m) = 0.924 J.
Since the work done by friction is negative (friction opposes motion), the kinetic energy at the end of the flat part of the track is given by KE = 4.75 J - 0.924 J = 3.826 J.
Finally, with the kinetic energy at the bottom of the right curved section equal to 3.826 J, we can find the height the block reaches using the equation KE = PE, where PE is the potential energy at the maximum height.
Using the equation PE = mgh, we can solve for h: h = KE / (mg) = (3.826 J) / (0.50 kg * 9.8 m/s^2) ≈ 0.78 m.
So, the maximum height reached by the block on the right curved portion of the track is approximately 0.78 meters.
Hope that lifts your spirits!
To find the maximum height reached by the block on the right curved portion of the track, we can use the principle of conservation of mechanical energy.
The total mechanical energy of the block is conserved, which means the sum of its potential energy and kinetic energy remains constant throughout its motion.
At the top of the left curved portion of the track, all of the potential energy is converted into kinetic energy since the block is released from rest. The potential energy at the top of the left curved portion of the track can be calculated using the formula:
Potential energy = m * g * h
where m is the mass of the block (0.50 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (96 cm = 0.96 m).
Potential energy = 0.50 kg * 9.8 m/s^2 * 0.96 m = 4.75 J
Since the curved portions of the track are frictionless, the block does not lose any energy due to friction. However, on the flat part of the track, there is kinetic friction that will convert some of the block's energy into heat.
The work done by friction on the block can be calculated using the formula:
Work done by friction = force of friction * distance
The force of friction can be calculated using the formula:
Force of friction = coefficient of kinetic friction * normal force
The normal force is equal to the weight of the block, which can be calculated using the formula:
Weight = m * g
where m is the mass of the block and g is the acceleration due to gravity.
The distance traveled on the flat part of the track is equal to the length of the flat part, L.
The work done by friction can be calculated as:
Work done by friction = (coefficient of kinetic friction * m * g) * L
The work done by friction can also be calculated as the difference in kinetic energy before and after the flat part of the track:
Work done by friction = final kinetic energy - initial kinetic energy
Since the block comes to a stop at the top of the right curved portion of the track, the final kinetic energy is zero. The initial kinetic energy is equal to the difference between the total mechanical energy and the potential energy at the top of the left curved portion of the track:
Initial kinetic energy = total mechanical energy - potential energy
The total mechanical energy at the start can be calculated as:
Total mechanical energy = potential energy at the top of the left curved portion of the track
Therefore:
Work done by friction = (total mechanical energy - potential energy) - 0
Substituting in the values:
Work done by friction = (4.75 J - 4.75 J) - 0 = 0 J
Since no work is done by friction, there is no loss of mechanical energy on the flat part of the track.
As a result, the maximum height reached by the block on the right curved portion of the track is equal to the initial potential energy at the top of the left curved portion of the track:
Maximum height = Potential energy = 4.75 J
Therefore, the maximum height reached by the block on the right curved portion of the track is 4.75 J.
To find the maximum height reached by the block on the right curved portion of the track, we can use the principle of conservation of mechanical energy. The mechanical energy of the block is conserved as it moves along the track, meaning the total energy at one point is equal to the total energy at another point.
First, let's calculate the initial mechanical energy of the block when it is released from rest from a height of 96 cm on the left curved portion of the track.
Potential energy (PE) = mgh
m = mass of the block = 0.50 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height = 96 cm = 0.96 m
PE = (0.50 kg)(9.8 m/s^2)(0.96 m)
PE = 4.75 J
Next, let's calculate the work done by friction as the block slides on the flat portion of the track.
Work done by friction (W) = force of friction * distance
Force of friction = coefficient of kinetic friction * normal force
Normal force = weight of the block = mg
Coefficient of kinetic friction (μ) = 0.103
W = (0.103)(0.50 kg)(9.8 m/s^2)(1.85 m)
W = 0.930 J (rounded to three decimal places)
Since the curved portions of the track are frictionless, the work done by friction only occurs on the flat portion of the track.
Finally, let's calculate the final mechanical energy of the block when it reaches the maximum height on the right curved portion of the track.
The final mechanical energy is equal to the sum of the potential energy and the work done by friction.
Final mechanical energy = PE + W
Final mechanical energy = 4.75 J + 0.930 J
Final mechanical energy = 5.68 J (rounded to three decimal places)
The final mechanical energy is equal to the potential energy at the maximum height on the right curved portion of the track.
Potential energy (PE) = mgh
Let's rearrange the equation and solve for h.
h = PE / (mg)
h = 5.68 J / (0.50 kg * 9.8 m/s^2)
h = 1.16 m (rounded to two decimal places)
Therefore, the maximum height reached by the block on the right curved portion of the track is approximately 1.16 meters.