Tag Archives: gravitational potential energy

College Physics by Openstax Chapter 7 Problem 3

(a) Calculate the work done on a 1500-kg elevator car by its cable to lift it 40.0 m at constant speed, assuming friction averages 100 N. (b) What is the work done on the elevator car by the gravitational force in this process? (c) What is the total work done on the elevator car?

Solution:

The work W that a force F does on an object is the product of the magnitude F of the force, times the magnitude d of the displacement, times the cosine of the angle \theta between them. In symbols,

W=Fd \cos \theta

Part A

The force in the cable is equal to the combined effect of the weight of the elevator and the friction that opposes the motion. That is

\begin{align*}
F & = mg + f \\
F & = \left( 1500\ \text{kg} \right)\left( 9.80\ \text{m/s}^2 \right)+100\ \text{N} \\
F & = 14800\ \text{N}
\end{align*}

This force in the cable is directed upward. The displacement is also upward, making the angle between the two quantities equal to zero. Thus, \theta = 0.

Substituting these values in the equation, the work done by the cable is

\begin{align*}
W & = Fd \cos \theta \\
W & = \left( 14\ 800\ \text{N} \right)\left( 40.0\ \text{m} \right) \cos 0^\circ \\
W & = 592\ 000\ \text{J} \\
W & = 5.92 \times 10^{5} \ \text{J} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part B

The force due to gravity is equal to the weight of the elevator alone. That is 

\begin{align*}
\text{Weight} & = mg \\
 & = \left( 1\ 500\ \text{kg} \right)\left( 9.80\ \text{m/s}^2 \right) \\
 & = 14\ 700\ \text{N}
\end{align*}

This force is directed downward, whereas the displacement is directed upward. Therefore, the angle \theta between the two quantities is \theta = 180^\circ.

Substituting these values in the formula for work, we have

\begin{align*}
W & = Fd \cos \theta \\
W & = \left( 14\ 700\ \text{N} \right)\left( 40.0\ \text{m} \right) \cos 180^\circ \\
W & = -588\ 000\ \text{J} \\
W & = -5.88 \times 10^{5}\ \text{J} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part C

Since the elevator is moving at a constant speed, it is in equilibrium. This means that the net external force experience by the elevator is zero. Therefore, the total work done on the elevator car is 

W_{T} = 0\ \text{J} \ \qquad \ \color{DarkOrange} \left( \text{Answer}  \right)

College Physics by Openstax Chapter 7 Problem 2

A 75.0-kg person climbs stairs, gaining 2.50 meters in height. Find the work done to accomplish this task. (Neglect friction in your calculations.)

Solution:

Work done against gravity in lifting an object becomes potential energy of the object-Earth system. The change in gravitational potential energy is \Delta PE_{g} = mgh, with h being the increase in height and g the acceleration due to gravity.

W=mgh

We are given the following values: m=75.0\ \text{kg}, g=9.80\ \text{m/s}^2, and h=2.50\ \text{m}.

Substitute the given in the formula.

\begin{align*}
W & = mgh \\
W & = \left( 75.0\ \text{kg} \right)\left( 9.80\ \text{m/s}^2 \right)\left( 2.50\ \text{m} \right)\\
W & = 1837.5\ \text{Nm} \\
W & = 1837.5\ \text{J} \\
W & = 1.84 \times 10 ^{3} \ \text{J} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

The work done is about 1.84 \times 10 ^ {3}\ \text{Joules} .

Solution Guides to College Physics by Openstax Chapter 7 Banner

Chapter 7: Work, Energy, and Energy Resources

Work: The Scientific Definition

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Kinetic Energy and the Work-Energy Theorem

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Gravitational Potential Energy

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Conservative Forces and Potential Energy

Problem 22

Problem 23

Nonconservative Forces

Problem 24

Problem 25

Conservation of Energy

Problem 26

Problem 27

Problem 28

Problem 29

Power

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Work, Energy, and Power in Humans

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

World Energy Use

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70