Capacitors, Magnetic Circuits, and Transformers is a free introductory textbook on the physics of capacitors, coils, and transformers. See the editorial for more information....

# Problems - Energy

 1-1 A body with a mass of 5 kg is elevated to a height of 100 m from a given solid surface and is then allowed to fall. Neglect air resistance and determine (a) The potential energy relative to the given surface when the body is 100 m above this surface. (b) The height at which the potential energy of this body equals the kinetic energy. (c) The velocity for the condition of part (b) above by equating the change in potential energy to the kinetic energy. (d) The irreversible energy for the condition of part (b) above if air resistance can be neglected. (e) The condition of the body and the surface such that the irreversible energy is zero after striking the given solid surface. 1-2 The body referred to in Problem 1 -1 is raised on an inclined surface to a height of 100m above a given solid surface. It is then allowed to slide down the inclined surface. At the instant the body has descended 50 m it has a velocity of 20 mps. When the body is at a height of 50 m, determine (a) The irreversible energy in the system, assuming that the amount of energy leaving the plane and the body to be negligible. (b) The stored energy that is reversible. 1-3 A 250-v, d-c motor delivers 10 hp at a speed of 600 rpm to a load. The voltage applied to the motor is 250 v and the current is 35.6 amp. Assume the motor to have been in operation for a sufficient period so that all its parts are at a steady temperature and determine (a) All the power into the motor. (b) All the power out of the motor. (c) The useful output. (d) The efficiency of the motor. (e) The output torque in newton meters. NOTE: 1 hp = 746 w. 1-4 A 3-phase, a-c generator delivers 100,000 kw at a speed of 3,600 rpm. The efficiency is 98.5 percent. Determine (a) The horsepower input. (b) The input torque in newton meters. (c) The power converted into heat. 1-5 The input to the voice coil of a loud-speaker is 200 mws. The acoustic output is 4 mw. Determine (a) The efficiency. (b) The power converted into heat. 1-6 When a constant d-c voltage V is applied to an inductive circuit of constant resistance R ohms and constant self-inductance of L h, the current is expressed by Where t is the time following the application of the voltage and e is the natural logarithmic base. Express as a function of time (a) The power input to this circuit. (b) The energy input to this system. (c) The irreversible energy. (d) The energy stored in the self-inductance. 1-7 The winding of an electromagnet has a resistance of 10 ohms and a self-inductance of 1.0 h. A constant d-c emf of 100 v is applied to this magnet. Determine (a) The initial current when the voltage is applied. (b) The final current. (c) The initial rate in amp per sec at which the current changes. (d) The time, after application of voltage, when the power that stores energy in the inductance is a maximum. (e) The maximum power taken by the inductance. (f) The time constant. 1-8 An inductive circuit has a resistance of 10 ohms and a self-inductance of 1 h. Determine the voltage applied to this circuit expressed as a function of time such that the current increases at a constant rate of 20 amp per sec. Assume i=0 at t=0. 1-9 When a constant d-c voltage is applied to a capacitive circuit having a resistance R ohms in series with an ideal capacitance C f, the current is expressed by Where t is the time following the application of the voltage, (a) Express as functions of time The power input to this circuit. The energy input to this circuit. The energy stored in the capacitance. The irreversible energy. (b) What is the ratio of irreversible energy to the stored energy as time t approaches infinity? (c) How much of the total supplied energy has been degraded by the time the capacitor is fully charged? Into what form is the energy degraded? 1-10 A constant d-c potential is applied to a series circuit having a resistance of 100 ohms and a capacitance of 1 μf. Determine (a) The time expressed in seconds after application of voltage when the power absorbed by the capacitor is a maximum, (b) Repeat part (a) above expressing the time in terms of the time constant T. (c) The amount of energy stored in the capacitor at the instant the power absorbed by the capacitor is a maximum if the voltage applied to the circuit is 600 v. 1-11 A flywheel in the form of a disc has a mass of 222 kg and a radius of 0.3 m. (a) Determine the kinetic energy of rotation when this flywheel is rotating at a speed of 954 rpm. (b) What is the average value of the power required to bring this flywheel from rest to a speed of 954 rpm in 1 min? (c) The duty cycle is such that the flywheel gives up energy during 0.2 sec and recovers energy during 1.8 sec. If the speed drops from 954 rpm to 944 rpm in the 0.2-sec period, what is the average value of the power delivered by the flywheel ? (d) What is the average value of the power delivered to the flywheel during the recovery period of 1.8 sec, if the speed is returned to 954 rpm? 1-12 A projectile with a mass of 5 kg has a velocity of 300 mps. Determine the kinetic energy in joules. 1-13 (a) Determine the value of a current flowing in a magnetic circuit having a self-inductance of 9 h such that the stored energy is the same as the kinetic energy in Problem 1-12. (b) Assume the magnetic circuit of (a) to be linear, i.e., the value of the self-inductance is constant for all values of current, and determine the inductive reactance for a frequency of 60 cps. (c) If the value of the current determined in (a) is the maximum instantaneous value of a sinusoidal 60-cycle current, what is the rms value of the applied voltage if the resistance of the circuit can be neglected? (d) What is the reactive power in kilovars consumed by the magnetic circuit under the above conditions (Kvar = I2 ω L x 10-3)? 1-14 A capacitor is rated 50 kvar, 4,160 v at 60 cycles. Determine (a) The capacitance (Kvar =E2ωL x 10-3). (b) The maximum instantaneous energy stored in the dielectric of the capacitor when operating under rated conditions. (c) The number of such capacitors that under rated conditions will store a maximum potential energy equal to the kinetic energy of the projectile of Problem 1-12. NOTE: The approximate weight of the capacitor mentioned above is 60 lb and the price is approximately \$160. 1-15 A self-inductance of L h and negligible resistance is connected in parallel with a noninductive resistance. At t = 0 a constant current of t is applied to the parallel combination. Express (a) The current through the inductance as a function of time. (b) The current through the resistance as a function of time. 1-16 In Problem 1-15 determine (a) The final energy stored in the self-inductance. (b) The total energy converted into heat. (c) The value of time at the instant the power supplied to the resistance is equal to the power supplied to the self-inductance. (d) The energy stored in the self-inductance when the power delivered to the resistance equals that delivered to the self-inductance. 1-17 Assume a 60-cell storage battery to have a constant terminal voltage of 120 v and to be rated 100 amp-hr. (Such a battery would be approximately equivalent to ten 12-v automobile storage batteries.) Determine (a) The electrical energy stored chemically in this battery. (b) The capacitance of a capacitor that would store the same amount of energy in its dielectric at a voltage of 120 v. (c) The kvar rating of this capacitor for 60-cycle operation if the maximum instantaneous applied voltage is 120 v. (d) The number of pounds of coal, having a Btu content of 14000, that have stored the same amount of energy. NOTE: 1 Btu = 1055j. 1-18 Determine the amount, expressed in feet, by which a spring, with a stiffness coefficient of 10 tons per inch, would be stretched to store the same amount of energy as the battery of Problem 1-17. How much force is required to stretch the spring to this extent? 1-19 (a) Express, as a function of time, the energy stored in the self-inductance L in R-L-C circuit, making use of Eq. 1-81. (b) Plot the energy stored in the self-inductance during the interval from bt = 0 to bt = 4π radians with a = 2b, where b is real. (c) Plot the current as a function of time during the interval from bt = 0 to bt = 4π radians with a = 2b, where b is real. 1-20 (a) Repeat Problem 1-19 when b is imaginary but a is real. (b) Determine the time after voltage is applied when the energy stored in the self-inductance is a maximum. 1-21 (a) Repeat Problem 1-19 when b is zero. HINT: sin bt approaches bt as bt approaches zero. (b) Determine the time after voltage is applied when the energy stored in the self-inductance is a maximum.

Last Update: 2011-01-12