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Integrated Concepts
(a) Assuming 95.0% efficiency for the conversion of electrical power by the motor, what current must the 12.0-V batteries of a 750-kg electric car be able to supply: (a) To accelerate from rest to 25.0 m/s in 1.00 min? (b) To climb a $2.00\times {\text{10}}^{\text{2}}\text{-m}$ -high hill in 2.00 min at a constant 25.0-m/s speed while exerting $5.00\times {\text{10}}^{\text{2}}\phantom{\rule{0.25em}{0ex}}\text{N}$ of force to overcome air resistance and friction? (c) To travel at a constant 25.0-m/s speed, exerting a $5.00\times {\text{10}}^{\text{2}}\phantom{\rule{0.25em}{0ex}}\text{N}$ force to overcome air resistance and friction? See [link] .
(a) 343 A
(b) $2\text{.}\text{17}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{A}$
(c) $1.10\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{A}$
Integrated Concepts
A light-rail commuter train draws 630 A of 650-V DC electricity when accelerating. (a) What is its power consumption rate in kilowatts? (b) How long does it take to reach 20.0 m/s starting from rest if its loaded mass is $5\text{.}\text{30}\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{kg}$ , assuming 95.0% efficiency and constant power? (c) Find its average acceleration. (d) Discuss how the acceleration you found for the light-rail train compares to what might be typical for an automobile.
Integrated Concepts
(a) An aluminum power transmission line has a resistance of $0\text{.}\text{0580}\phantom{\rule{0.25em}{0ex}}\Omega /\text{km}$ . What is its mass per kilometer? (b) What is the mass per kilometer of a copper line having the same resistance? A lower resistance would shorten the heating time. Discuss the practical limits to speeding the heating by lowering the resistance.
(a) $1.23\times {\text{10}}^{\text{3}}\phantom{\rule{0.25em}{0ex}}\text{kg}$
(b) $2.64\times {\text{10}}^{\text{3}}\phantom{\rule{0.25em}{0ex}}\text{kg}$
Integrated Concepts
(a) An immersion heater utilizing 120 V can raise the temperature of a $1.00\times {\text{10}}^{\text{2}}\text{-g}$ aluminum cup containing 350 g of water from $\text{20}\text{.}\mathrm{0\xba}\text{C}$ to $\text{95}\text{.}\mathrm{0\xba}\text{C}$ in 2.00 min. Find its resistance, assuming it is constant during the process. (b) A lower resistance would shorten the heating time. Discuss the practical limits to speeding the heating by lowering the resistance.
Integrated Concepts
(a) What is the cost of heating a hot tub containing 1500 kg of water from $\text{10}\text{.}\mathrm{0\xba}\text{C}$ to $\text{40}\text{.}\mathrm{0\xba}\text{C}$ , assuming 75.0% efficiency to account for heat transfer to the surroundings? The cost of electricity is $\text{9}\phantom{\rule{0.25em}{0ex}}\text{cents/kW}\cdot \text{h}$ . (b) What current was used by the 220-V AC electric heater, if this took 4.00 h?
Unreasonable Results
(a) What current is needed to transmit $1.00\times {\text{10}}^{\text{2}}\phantom{\rule{0.25em}{0ex}}\text{MW}$ of power at 480 V? (b) What power is dissipated by the transmission lines if they have a $1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{-}\phantom{\rule{0.25em}{0ex}}\Omega $ resistance? (c) What is unreasonable about this result? (d) Which assumptions are unreasonable, or which premises are inconsistent?
(a) $2.08\times {\text{10}}^{\text{5}}\phantom{\rule{0.25em}{0ex}}\text{A}$
(b) $4.33\times {\text{10}}^{\text{4}}\phantom{\rule{0.25em}{0ex}}\text{MW}$
(c) The transmission lines dissipate more power than they are supposed to transmit.
(d) A voltage of 480 V is unreasonably low for a transmission voltage. Long-distance transmission lines are kept at much higher voltages (often hundreds of kilovolts) to reduce power losses.
Unreasonable Results
(a) What current is needed to transmit $1.00\times {\text{10}}^{\text{2}}\phantom{\rule{0.25em}{0ex}}\text{MW}$ of power at 10.0 kV? (b) Find the resistance of 1.00 km of wire that would cause a 0.0100% power loss. (c) What is the diameter of a 1.00-km-long copper wire having this resistance? (d) What is unreasonable about these results? (e) Which assumptions are unreasonable, or which premises are inconsistent?
Construct Your Own Problem
Consider an electric immersion heater used to heat a cup of water to make tea. Construct a problem in which you calculate the needed resistance of the heater so that it increases the temperature of the water and cup in a reasonable amount of time. Also calculate the cost of the electrical energy used in your process. Among the things to be considered are the voltage used, the masses and heat capacities involved, heat losses, and the time over which the heating takes place. Your instructor may wish for you to consider a thermal safety switch (perhaps bimetallic) that will halt the process before damaging temperatures are reached in the immersion unit.
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