8) A simple pendulum with length 2.23 m and a mass of 6.74 kg is given an initial speed of 2.06 m/s at its equilibrium position. Assume it undergoes simple harmonic motion. Determine a) it's period, b) its total energy, c) its maximum angular displacement. d) Write an equation for the angular position at future times based on this information. (a: 3.00 s, b: 14.3 J, c: 0.441 rad, d: e (t)=0.441 rad sin(2.10s¯†t) )
8) A simple pendulum with length 2.23 m and a mass of 6.74 kg is given an initial speed of 2.06 m/s at its equilibrium position. Assume it undergoes simple harmonic motion. Determine a) it's period, b) its total energy, c) its maximum angular displacement. d) Write an equation for the angular position at future times based on this information. (a: 3.00 s, b: 14.3 J, c: 0.441 rad, d: e (t)=0.441 rad sin(2.10s¯†t) )
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
Related questions
Concept explainers
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
Topic Video
Question
![### Simple Pendulum and Simple Harmonic Motion
A simple pendulum with a length of 2.23 meters and a mass of 6.74 kg is given an initial speed of 2.06 meters per second (m/s) at its equilibrium position. The pendulum undergoes simple harmonic motion.
#### Problems to Solve:
1. **Period (T)**:
- The time taken for one complete cycle of the pendulum's swing.
2. **Total Energy (E)**:
- The sum of the kinetic and potential energy in the system.
3. **Maximum Angular Displacement (θ_max)**:
- The furthest angle from the equilibrium position the pendulum reaches.
4. **Equation for Angular Position as a Function of Time**:
- This expresses the pendulum's angular displacement over time.
#### Solutions:
- **Period (T)**:
\( T = 3.00 \, \text{s} \)
- **Total Energy (E)**:
\( E = 14.3 \, \text{J} \)
- **Maximum Angular Displacement (θ_max)**:
\( \theta_{\text{max}} = 0.441 \, \text{rad} \)
- **Equation for Angular Position (θ)**:
\[
\theta(t) = 0.441 \, \text{rad} \sin(2.10 \, \text{s}^{-1} t)
\]
The equation above shows that the pendulum's angular position oscillates as a sine function, where:
- \( 0.441 \, \text{rad} \) is the amplitude,
- \( 2.10 \, \text{s}^{-1} \) is the angular frequency,
- \( t \) is the time variable.
### Explanation of Key Terms:
- **Simple Harmonic Motion**:
- This is the type of oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement.
- **Angular Displacement (θ)**:
- The angle through which the pendulum has swung from its equilibrium (rest) position.
- **Amplitude**:
- The maximum displacement from the equilibrium position in simple harmonic motion.
- **Angular Frequency (ω)**:
- Related to the period, it describes how many oscillations occur in a unit of time.
This model assumes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45c3f407-ef34-4ad6-a068-fef4425d0691%2F77859b92-7a05-4121-a74d-a5498fd41be0%2F7wge75p_processed.png&w=3840&q=75)
Transcribed Image Text:### Simple Pendulum and Simple Harmonic Motion
A simple pendulum with a length of 2.23 meters and a mass of 6.74 kg is given an initial speed of 2.06 meters per second (m/s) at its equilibrium position. The pendulum undergoes simple harmonic motion.
#### Problems to Solve:
1. **Period (T)**:
- The time taken for one complete cycle of the pendulum's swing.
2. **Total Energy (E)**:
- The sum of the kinetic and potential energy in the system.
3. **Maximum Angular Displacement (θ_max)**:
- The furthest angle from the equilibrium position the pendulum reaches.
4. **Equation for Angular Position as a Function of Time**:
- This expresses the pendulum's angular displacement over time.
#### Solutions:
- **Period (T)**:
\( T = 3.00 \, \text{s} \)
- **Total Energy (E)**:
\( E = 14.3 \, \text{J} \)
- **Maximum Angular Displacement (θ_max)**:
\( \theta_{\text{max}} = 0.441 \, \text{rad} \)
- **Equation for Angular Position (θ)**:
\[
\theta(t) = 0.441 \, \text{rad} \sin(2.10 \, \text{s}^{-1} t)
\]
The equation above shows that the pendulum's angular position oscillates as a sine function, where:
- \( 0.441 \, \text{rad} \) is the amplitude,
- \( 2.10 \, \text{s}^{-1} \) is the angular frequency,
- \( t \) is the time variable.
### Explanation of Key Terms:
- **Simple Harmonic Motion**:
- This is the type of oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement.
- **Angular Displacement (θ)**:
- The angle through which the pendulum has swung from its equilibrium (rest) position.
- **Amplitude**:
- The maximum displacement from the equilibrium position in simple harmonic motion.
- **Angular Frequency (ω)**:
- Related to the period, it describes how many oscillations occur in a unit of time.
This model assumes
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON