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A flywheel is fitted to the crankshaft of an engine having W as the amount of indicated work per revolution and permissible limits of coefficient of fluctuation of energy and speed as CEand CS respectively. The kinetic energy of the flywheel is given by
- a)2WCE/CS
- b)WCE/2CS
- c)WCE/CS
- d)WCS/2CE
Correct answer is option 'B'. Can you explain this answer?
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A flywheel is fitted to the crankshaft of an engine having W as the am...
Explanation:
Flywheel:
A flywheel is a mechanical device that is designed to store energy in the form of rotational kinetic energy. It is fitted to the crankshaft of an engine to maintain a constant speed and reduce fluctuations in speed.
Coefficient of Fluctuation of Energy (CE):
The coefficient of fluctuation of energy is defined as the ratio of the maximum fluctuation of energy to the mean energy supplied per cycle.
CE = (Emax - Emin) / (Emax + Emin)
where Emax = maximum energy and Emin = minimum energy.
Permissible limit of CE is 0.05 or 5%.
Coefficient of Fluctuation of Speed (CS):
The coefficient of fluctuation of speed is defined as the ratio of the maximum fluctuation of speed to the mean speed.
CS = (Vmax - Vmin) / (Vmax + Vmin)
where Vmax = maximum speed and Vmin = minimum speed.
Permissible limit of CS is 0.1 or 10%.
Kinetic energy of the flywheel:
The kinetic energy of the flywheel is given by the formula:
KE = (1/2) I ω^2
where I = moment of inertia of the flywheel and ω = angular velocity of the flywheel.
We know that the work done by the engine per revolution is W.
Therefore, the energy stored in the flywheel per revolution is also W.
Equating the two, we get:
KE = W
Substituting the value of KE in the formula, we get:
(1/2) I ω^2 = W
ω^2 = 2W/I
ω = √(2W/I)
Substituting the value of ω in the formula of KE, we get:
KE = (1/2) I (2W/I)
KE = W
Hence, the kinetic energy of the flywheel is equal to the work done per revolution.
Calculation of CE and CS:
The maximum and minimum energy and speed can be calculated as follows:
Emax = W + (1/2) I ω^2
Emin = W - (1/2) I ω^2
Vmax = ω (1+ CS/2)
Vmin = ω (1- CS/2)
Substituting the value of ω in the above equations, we get:
Emax = W + (1/2) I (2W/I)
Emax = 3/2 W
Emin = W - (1/2) I (2W/I)
Emin = 1/2 W
Vmax = √(2W/I) (1+ CS/2)
Vmin = √(2W/I) (1- CS/2)
Substituting the permissible limits of CE and CS, we get:
0.05 = (Emax - Emin) / (Emax + Emin)
0.05 = (3/2 W - 1/2 W) / (3/2 W + 1/2 W)
0.05 = 1/4
Therefore, the value of CE is 0.05 or 5%.
0.1 = (Vmax - Vmin) / (Vmax + Vmin)
0.1 = (√(2W/I) (1+ CS/2) -
Flywheel:
A flywheel is a mechanical device that is designed to store energy in the form of rotational kinetic energy. It is fitted to the crankshaft of an engine to maintain a constant speed and reduce fluctuations in speed.
Coefficient of Fluctuation of Energy (CE):
The coefficient of fluctuation of energy is defined as the ratio of the maximum fluctuation of energy to the mean energy supplied per cycle.
CE = (Emax - Emin) / (Emax + Emin)
where Emax = maximum energy and Emin = minimum energy.
Permissible limit of CE is 0.05 or 5%.
Coefficient of Fluctuation of Speed (CS):
The coefficient of fluctuation of speed is defined as the ratio of the maximum fluctuation of speed to the mean speed.
CS = (Vmax - Vmin) / (Vmax + Vmin)
where Vmax = maximum speed and Vmin = minimum speed.
Permissible limit of CS is 0.1 or 10%.
Kinetic energy of the flywheel:
The kinetic energy of the flywheel is given by the formula:
KE = (1/2) I ω^2
where I = moment of inertia of the flywheel and ω = angular velocity of the flywheel.
We know that the work done by the engine per revolution is W.
Therefore, the energy stored in the flywheel per revolution is also W.
Equating the two, we get:
KE = W
Substituting the value of KE in the formula, we get:
(1/2) I ω^2 = W
ω^2 = 2W/I
ω = √(2W/I)
Substituting the value of ω in the formula of KE, we get:
KE = (1/2) I (2W/I)
KE = W
Hence, the kinetic energy of the flywheel is equal to the work done per revolution.
Calculation of CE and CS:
The maximum and minimum energy and speed can be calculated as follows:
Emax = W + (1/2) I ω^2
Emin = W - (1/2) I ω^2
Vmax = ω (1+ CS/2)
Vmin = ω (1- CS/2)
Substituting the value of ω in the above equations, we get:
Emax = W + (1/2) I (2W/I)
Emax = 3/2 W
Emin = W - (1/2) I (2W/I)
Emin = 1/2 W
Vmax = √(2W/I) (1+ CS/2)
Vmin = √(2W/I) (1- CS/2)
Substituting the permissible limits of CE and CS, we get:
0.05 = (Emax - Emin) / (Emax + Emin)
0.05 = (3/2 W - 1/2 W) / (3/2 W + 1/2 W)
0.05 = 1/4
Therefore, the value of CE is 0.05 or 5%.
0.1 = (Vmax - Vmin) / (Vmax + Vmin)
0.1 = (√(2W/I) (1+ CS/2) -
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