Resnick & Halliday Solutions for Chapter: Kinetic Energy and Work, Exercise 1: Problems

Across a horizontal floor, a $102 \mathrm{kg}$ block is pulled at a constant speed of $5.5 \mathrm{m} {\mathrm{s}}^{-1}$ by an applied force of $125 \mathrm{N}$ directed $38°$ above the horizontal. Calculate the rate at which the force does work on the block.($\cos \left(38°\right)=0.788$)

The loaded cab of an elevator has a mass of $5.0\times {10}^3 \mathrm{kg}$ and moves $210 \mathrm{m}$ up the shaft in $23 \mathrm{s}$ at constant speed. At what average rate does the force from the cable does work on the cab?

A machine carries a $4.0 \mathrm{kg}$ package from an initial position of ${\overrightarrow{d}}_{\mathrm{i}}=\left(0.50 \mathrm{m}\right)\hat{i}+\left(0.75 \mathrm{m}\right)\hat{j}+\left(0.20 \mathrm{m}\right)\hat{k}$ at $t=0$ to a final position of ${\overrightarrow{d}}_{\mathrm{f}}=\left(7.50 \mathrm{m}\right)\hat{i}+\left(12.0 \mathrm{m}\right)\hat{j}+\left(7.20 \mathrm{m}\right)\hat{k}$ at $t=12 \mathrm{s}$. The constant force applied by the machine on the package is $\overrightarrow{F}=\left(2.00 \mathrm{N}\right)\hat{i}+\left(4.00 \mathrm{N}\right)\hat{j}+\left(6.00 \mathrm{N}\right)\hat{k}$. For that displacement find (a) the work done on the package by the machine's force and (b) the average power of the machine's force on the package.

A $0.35 \mathrm{kg}$ ladle sliding on a horizontal frictionless surface is attached to one end of a horizontal spring $\left(\mathrm{k}=500 \mathrm{N} {\mathrm{m}}^{-1}\right)$ whose other end is fixed. The ladle has a kinetic energy of $10 \mathrm{J}$ as it passes through its equilibrium position (the point at which the spring force is zero). (a) At what rate is the spring doing work on the ladle as the ladle passes through its equilibrium position?
(b) At what rate is the spring doing work on the ladle when the spring is compressed $0.10 \mathrm{m}$ and the ladle is moving away from the equilibrium position?

A fully loaded, slow-moving freight elevator has a cab with a total mass of $1200 \mathrm{kg}$, which is required to travel upward $54 \mathrm{m}$ in $3.0 \min$, starting and ending at rest. The elevator's counterweight has a mass of only $950 \mathrm{kg}$, and so the elevator motor must help. What average power is required of the force the motor exerts on the cab via the cable?

(a) At a certain instant, a particle-like object is acted on by a force $\overrightarrow{F}=\left(4.0 \mathrm{N}\right)\hat{\mathrm{i}}-\left(2.0 \mathrm{N}\right)\hat{\mathrm{j}}+\left(9.0 \mathrm{N}\right)\hat{\mathrm{k}}$, while the object's velocity is $\overrightarrow{v}=-\left(2.0 \mathrm{m} {\mathrm{s}}^{-1}\right)\hat{\mathrm{i}}+\left(4.0 \mathrm{m} {\mathrm{s}}^{-1}\right)\hat{\mathrm{k}}.$ What is the instantaneous rate at which the force does work on the object?
(b) At some other time, the velocity consists of only a $y$ component. If the force is unchanged and the instantaneous power is $-15 \mathrm{W}$, what is the velocity of the object?

A force $\overrightarrow{F}=\left(3.00 \mathrm{N}\right)\hat{\mathrm{i}}+\left(7.00 \mathrm{N}\right)\hat{\mathrm{j}}+\left(7.00 \mathrm{N}\right)\hat{\mathrm{k}}$ acts on a $2.00 \mathrm{kg}$ mobile object that moves from an initial position of ${\overrightarrow{d}}_{\mathrm{i}}=\left(3.00 \mathrm{m}\right)\hat{\mathrm{i}}-\left(2.00 \mathrm{m}\right)\hat{\mathrm{j}}+\left(5.00 \mathrm{m}\right)\hat{\mathrm{k}}$ to a final position of ${\overrightarrow{d}}_{\mathrm{f}}=-\left(5.00 \mathrm{m}\right)\hat{\mathrm{i}}+\left(4.00\mathrm{m}\right)\hat{\mathrm{j}}+\left(7.00 \mathrm{m}\right)\hat{\mathrm{k}}$ in $4.00 \mathrm{s}.$ Find,
(a) the work done on the object by the force in the $4.00 \mathrm{s}$ interval,
(b) the average power due to the force during that interval and
(c) the angle between the vectors ${\overrightarrow{d}}_{\mathrm{i}}$ and ${\overrightarrow{d}}_{\mathrm{f}}$.

A funny car accelerates from rest through a measured track distance in time $T$ with the engine operating at constant power $P$. If the track crew can increase the engine power by a differential amount $\mathrm{d}P$, what is the change in the time required for the run?