Physics
Physics
1st Edition
Walker
ISBN: 9780133256925
Table of contents
Textbook solutions

All Solutions

Page 459: Practice Problems

Exercise 8
Step 1
1 of 3
### Theoretical reminder

For a mass $m$ attached to a spring with a spring constant $k$ we know that the period of simple harmonic oscillations will be:

$$
begin{equation}
T = 2 , pi ; sqrt{frac{m}{k}}
end{equation}
$$

Step 2
2 of 3
### Calculation

In our problem the mass attached to the spring increases by a factor of 4, so we can write for the old and new period:

$$
begin{align*}
T_1 &= 2 , pi ; sqrt{frac{m_1}{k}} \
T_2 &= 2 , pi ; sqrt{frac{m_2}{k}}
end{align*}
$$

Now inserting $m_2 = 4 , m_1$ we have:

$$
begin{align*}
T_2 = 2 , pi ; sqrt{frac{4 , m_1}{k}} = sqrt{4} cdot 2 , pi ; sqrt{frac{m_1}{k}} = 2 cdot T_1
end{align*}
$$

Result
3 of 3
The period increases by a factor of 2, that is $T_2 = 2 , T_1$
Exercise 9
Step 1
1 of 3
### Theoretical reminder

For a mass $m$ attached to a spring with a spring constant $k$ we know that the period of simple harmonic oscillations will be:

$$
begin{equation}
T = 2 , pi ; sqrt{frac{m}{k}}
end{equation}
$$

Step 2
2 of 3
### Calculation

In our problem both the mass attached to the spring and the spring constant increases by a factor of 2, so we can write:

For the old and new period:

$$
begin{align*}
T_1 &= 2 , pi ; sqrt{frac{m_1}{k_1}} \
T_2 &= 2 , pi ; sqrt{frac{m_2}{k_1}}
end{align*}
$$

Now inserting $m_2 = 2 , m_1$ and $k_2 = 2 , k_1$ we have:

$$
begin{align*}
T_2 = 2 , pi ; sqrt{frac{2 , m_1}{2k_1}} = 2pi ; sqrt{frac{m_1}{k}} = T_1
end{align*}
$$

Result
3 of 3
When both the mass and the spring constant are doubled the period remains the same, that is $T_2 = T_1$
Exercise 10
Step 1
1 of 4
### Theoretical reminder

For a mass $m$ attached to a spring with a spring constant $k$ we know that the period of simple harmonic oscillations will be:

$$
begin{equation}
T = 2 , pi ; sqrt{frac{m}{k}}
end{equation}
$$

We know that period $T$ is defined as the time required to complete one full cycle of a given motion.

The frequency $f$ is the number of oscillations per unit time.

They are related by the following formula:

$$
begin{equation}
f = frac{1}{T}
end{equation}
$$

Step 2
2 of 4
### Knowns

The mass attached to our spring is $m = 0.32text{ kg}$

The frequency of our harmonic motion $f = 1.6text{ Hz}$

Step 3
3 of 4
### Calculation

First we will use the frequency-period relation (2) to find the period of our motion:

$$
begin{align*}
T = frac{1}{f} = frac{1}{1.6text{ Hz}} = 0.625text{ s}
end{align*}
$$

Now we will use formula (1) to find the spring constant $k$ using the mass of the spring, and the period of the motion.

Rearranging for the spring constant we have:

$$
begin{align*}
T &= 2 , pi ; sqrt{frac{m}{k}} \
T^2 &= left(2 , pi right)^2 ; frac{m}{k} \
k &= frac{4 , pi^2}{T^2} ; m
end{align*}
$$

Plugging in the values we get:

$$
begin{align*}
k = frac{4 , pi^2}{(0.625text{ s})^2 } cdot 0.32text{ kg} = 32.34 ; frac{text{N}}{text{m}}
end{align*}
$$

Result
4 of 4
The spring constant is $k = 32.34 ; frac{text{N}}{text{m}}$
Exercise 11
Step 1
1 of 4
### Theoretical reminder

For a mass $m$ attached to a spring with a spring constant $k$ we know that the period of simple harmonic oscillations will be:

$$
begin{equation}
T = 2 , pi ; sqrt{frac{m}{k}}
end{equation}
$$

Step 2
2 of 4
### Knowns

The spring constant of our spring is $k = 22 ; frac{text{N}}{text{m}}$

The period of our harmonic motion $T = 0.95text{ s}$

Step 3
3 of 4
### Calculation

We will use formula (1) to find the mass $m$ of our spring using the spring constant $k$ of the spring, and the period of the motion.

Rearranging for the mass we have:

$$
begin{align*}
T &= 2 , pi ; sqrt{frac{m}{k}} \
T^2 &= left(2 , pi right)^2 ; frac{m}{k} \
m &= frac{T^2}{(2 , pi)^2} ; k
end{align*}
$$

Plugging in the values we get:

$$
begin{align*}
m = frac{(0.95text{ s})^2 }{4 , pi^2 } cdot 22 ; frac{text{N}}{text{m}} approx 0.50text{ kg}
end{align*}
$$

Result
4 of 4
The mass of the spring is $m = 0.50text{ kg}$
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Chapter 1: Introduction to Physics
Section 1.1: Physics and the Scientific Method
Section 1.2: Physics and Society
Section 1.3: Units and Dimensions
Section 1.4: Basic Math for Physics
Page 38: Assessment
Page 41: Standardized Test Prep
Chapter 2: Introduction to Motion
Section 2.1: Describing Motion
Section 2.2: Speed and Velocity
Section 2.3: Position-Time Graphs
Section 2.4: Equation of Motion
Page 66: Assessment
Page 71: Standardized Test Prep
Page 45: Practice Problems
Page 47: Practice Problems
Page 47: Lesson Check
Page 49: Practice Problems
Page 52: Practice Problems
Page 53: Lesson Check
Page 56: Practice Problems
Page 57: Lesson Check
Page 59: Practice Problems
Page 60: Practice Problems
Page 62: Practice Problems
Page 62: Lesson Check
Chapter 3: Acceleration and Acceleration Motion
Section 3.1: Acceleration
Section 3.2: Motion with Constant Acceleration
Section 3.3: Position-Time Graphs for Constant Acceleration
Section 3.4: Free Fall
Page 105: Assessment
Page 111: Standardized Test Prep
Chapter 4: Motion in Two Dimensions
Section 4.1: Vectors in Physics
Section 4.2: Adding and Subtracting Vectors
Section 4.3: Relative Motion
Section 4.4: Projectile Motion
Page 144: Assessment
Page 149: Standardized Test Prep
Chapter 5: Newton’s Laws of Motion
Section 5.1: Newton’s Laws of Motion
Section 5.2: Applying Newton’s Laws
Section 5.3: Friction
Page 180: Assessment
Page 187: Standardized Test Prep
Chapter 6: Work and Energy
Section 6.1: Work
Section 6.2: Work and Energy
Section 6.3: Conservation of Energy
Section 6.4: Power
Page 220: Assessment
Page 227: Standardized Test Prep
Page 191: Practice Problems
Page 193: Practice Problems
Page 196: Lesson Check
Page 196: Practice Problems
Page 199: Practice Problems
Page 201: Practice Problems
Page 203: Practice Problems
Page 204: Practice Problems
Page 205: Practice Problems
Page 206: Lesson Check
Page 209: Practice Problems
Page 211: Lesson Check
Page 213: Practice Problems
Page 214: Practice Problems
Page 215: Practice Problems
Page 216: Lesson Check
Chapter 7: Linear Momentum and Collisions
Section 7.1: Momentum
Section 7.2: Impulse
Section 7.3: Conservation of Momentum
Section 7.4: Collisions
Page 260: Assessment
Page 265: Standardized Test Prep
Chapter 8: Rotational Motion and Equilibrium
Section 8.1: Describing Angular Motion
Section 8.2: Rolling Motion and the Moment of Inertia
Section 8.3: Torque
Section 8.4: Static Equilibrium
Page 300: Assessment
Page 305: Standardized Test Prep
Page 269: Practice Problems
Page 271: Practice Problems
Page 272: Practice Problems
Page 275: Practice Problems
Page 275: Lesson Check
Page 277: Practice Problems
Page 280: Lesson Check
Page 284: Practice Problems
Page 286: Practice Problems
Page 287: Practice Problems
Page 289: Lesson Check
Page 294: Practice Problems
Page 295: Practice Problems
Page 296: Lesson Check
Chapter 9: Gravity and Circular Motion
Section 9.1: Newton’s Law of Universal Gravity
Section 9.2: Applications of Gravity
Section 9.3: Circular Motion
Section 9.4: Planetary Motion and Orbits
Page 336: Assessment
Page 341: Standardized Test Prep
Chapter 10: Temperature and Heat
Section 10.1: Temperature, Energy, and Heat
Section 10.2: Thermal Expansion and Energy Transfer
Section 10.3: Heat Capacity
Section 10.4: Phase Changes and Latent Heat
Page 378: Assessment
Page 383: Standardized Test Prep
Chapter 11: Thermodynamics
Section 11.1: The First Law of Thermodynamics
Section 11.2: Thermal Processes
Section 11.3: The Second and Third Laws of Thermodynamics
Page 410: Assessment
Page 413: Standardized Test Prep
Chapter 12: Gases, Liquids, and Solids
Section 12.1: Gases
Section 12.2: Fluids at Rest
Section 12.3: Fluids in Motion
Section 12.4: Solids
Page 446: Assessment
Page 451: Standardized Test Prep
Chapter 13: Oscillations and Waves
Section 13.1: Oscillations and Periodic Motion
Section 13.2: The Pendulum
Section 13.3: Waves and Wave Properties
Section 13.4: Interacting Waves
Page 486: Assessment
Page 491: Standardized Test Prep
Chapter 14: Sound
Section 14.1: Sound Waves and Beats
Section 14.2: Standing Sound Waves
Section 14.3: The Doppler Effect
Section 14.4: Human Perception of Sound
Page 523: Assessment
Page 527: Standardized Test Prep
Page 495: Practice Problems
Page 496: Practice Problems
Page 500: Practice Problems
Page 501: Lesson Check
Page 503: Practice Problems
Page 504: Practice Problems
Page 506: Practice Problems
Page 506: Lesson Check
Page 510: Practice Problems
Page 511: Practice Problems
Page 512: Lesson Check
Page 514: Practice Problems
Page 516: Practice Problems
Page 517: Practice Problems
Page 519: Lesson Check
Chapter 15: The Properties of Lights
Section 15.1: The Nature of Light
Section 15.2: Color and the Electromagnetic Spectrum
Section 15.3: Polarization and Scattering of Light
Page 557: Assessment
Page 563: Standardized Test Prep
Chapter 16: Reflection and Mirrors
Section 16.1: The Reflection of Light
Section 16.2: Plane Mirrors
Section 16.3: Curved Mirrors
Page 590: Assessment
Page 595: Standardized Test Prep
Chapter 17: Refraction and Lenses
Section 17.1: Refraction
Section 17.2: Applications of Refraction
Section 17.3: Lenses
Section 17.4: Applications of Lenses
Page 629: Assessment
Page 635: Standardized Test Prep
Chapter 18: Interference and Diffraction
Section 18.1: Interference
Section 18.2: Interference in Thin Films
Section 18.3: Diffraction
Section 18.4: Diffraction Gratings
Page 668: Assessment
Page 673: Standardized Test Prep
Chapter 19: Electric Charges and Forces
Section 19.1: Electric Charge
Section 19.2: Electric Force
Section 19.3: Combining Electric Forces
Page 698: Assessment
Page 703: Standardized Test Prep
Chapter 20: Electric Fields and Electric Energy
Section 20.1: The Electric Field
Section 20.2: Electric Potential Energy and Electric Potential
Section 20.3: Capacitance and Energy Storage
Page 738: Assessment
Page 743: Standardized Test Prep
Chapter 21: Electric Current and Electric Circuits
Section 21.1: Electric Current, Resistance, and Semiconductors
Section 21.2: Electric Circuits
Section 21.3: Power and Energy in Electric Circuits
Page 775: Assessment
Page 781: Standardized Test Prep
Chapter 22: Magnetism and Magnetic Fields
Section 22.1: Magnets and Magnetic Fields
Section 22.2: Magnetism and Electric Currents
Section 22.3: The Magnetic Force
Page 810: Assessment
Page 815: Standardized Test Prep
Chapter 23: Electromagnetic Induction
Section 23.1: Electricity from Magnetism
Section 23.2: Electric Generators and Motors
Section 23.3: AC Circuits and Transformers
Page 844: Assessment
Page 849: Standardized Test Prep
Chapter 24: Quantum Physics
Section 24.1: Quantized Energy and Photons
Section 24.2: Wave-Particle Duality
Section 24.3: The Heisenberg Uncertainty Principle
Page 876: Assessment
Page 881: Standardized Test Prep
Chapter 26: Nuclear Physics
Section 26.1: The Nucleus
Section 26.2: Radioactivity
Section 26.3: Applications of Nuclear Physics
Section 26.4: Fundamental Forces and Elementary Particles
Page 944: Assessment
Page 947: Standardized Test Prep