An LC circuit consists of a 3.4-µF capacitor and a coil with a self-inductance 0.080 H and no appreciable resistance. At t = 0 t
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1. 0.42 Hz
The frequency of a simple harmonic motion for a spring is given by:
where
k = 7 N/m is the spring constant
m = 1 kg is the mass attached to the spring
Substituting these numbers into the formula, we find
2. 2.38 s
The period of the harmonic motion is equal to the reciprocal of the frequency:
where f = 0.42 Hz is the frequency. Substituting into the formula, we find
3. 0.4 m
The amplitude in a simple harmonic motion corresponds to the maximum displacement of the mass-spring system. In this case, the mass is initially displaced by 0.4 m: this means that during its oscillation later, the displacement cannot be larger than this value (otherwise energy conservation would be violated). Therefore, this represents the maximum displacement of the mass-spring system, so it corresponds to the amplitude.
4. 0.19 m
We can solve this part of the problem by using the law of conservation of energy. In fact:
- When the mass is released from equilibrium position, the compression/stretching of the spring is zero: , so the elastic potential energy is zero, and all the mechanical energy of the system is just equal to the kinetic energy of the mass:
where m = 1 kg and v = 0.5 m/s is the initial velocity of the mass
- When the spring reaches the maximum compression/stretching (x=A=amplitude), the velocity of the system is zero, so the kinetic energy is zero, and all the mechanical energy is just elastic potential energy:
Since the total energy must be conserved, we have:
5. Amplitude of the motion: 0.44 m
We can use again the law of conservation of energy.
- is the initial mechanical energy of the system, with
being the initial displacement of the mass and
being the initial velocity
is the mechanical energy of the system when x=A (maximum displacement)
Equalizing the two expressions, we can solve to find A, the amplitude:
6. Maximum velocity: 1.17 m/s
We can use again the law of conservation of energy.
- is the initial mechanical energy of the system, with
being the initial displacement of the mass and
being the initial velocity
is the mechanical energy of the system when x=0, which is when the system has maximum velocity,
Equalizing the two expressions, we can solve to find , the maximum velocity:
(a) The efficiency of an engine is defined as the ratio between the work done by the engine and the heat it takes in:
The engine in this problem does a work of
and it takes in 
of heat, therefore its efficiency is
(b) The heat taken by the machine is 4000 J; of this amount of heat, only 1100 J are converted into useful work. This means that the rest of the heat is wasted. Therefore, the wasted heat is the difference between the heat in input and the work done by the engine:
The engine in this problem does a work of
(b) The heat taken by the machine is 4000 J; of this amount of heat, only 1100 J are converted into useful work. This means that the rest of the heat is wasted. Therefore, the wasted heat is the difference between the heat in input and the work done by the engine: