Questions and Answers
How is the elastic constant (k) related to the restoring force (F) and displacement (s)?
The elastic constant (k) is the proportionality constant in Hooke’s law, relating the restoring force (F) to the displacement (s) from the equilibrium position as F = k * s.
What is the formula to calculate the displacement (s) when a spring is stretched or compressed?
The displacement (s) is calculated as s = l - l_0, where l is the new length and l_0 is the original length of the spring.
In the context of SHM, what happens to the velocity and acceleration when the displacement is maximal?
When the displacement is maximal, the velocity is zero while the acceleration is maximal, directed towards the equilibrium position.
Explain the significance of the negative sign in the relation a ∝ -s in SHM.
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What is the SI unit of the elastic constant (k), and why is it important?
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What is the SI unit of amplitude in simple harmonic motion?
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If the frequency of a simple harmonic oscillator is increased, what happens to its period?
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What is the relationship between the period of a pendulum and its length?
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How is the frequency (f) of an oscillator calculated from its period (T)?
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In calculating the period of a simple pendulum, what effect does increasing the acceleration due to gravity have?
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What is the correct formula for calculating the elastic constant (k) of a spring?
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What is the angular frequency (ω) of an oscillator related to its elastic constant (k) and mass (m)?
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Which variable can be plotted on the y-axis to analyze the relationship between the length of a pendulum and its period squared?
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What precautions should be taken during the pendulum experiment to ensure valid results?
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If a pendulum's length is doubled, what effect does this have on its period?
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What is the relationship between force (F) and displacement (s) according to Hooke's Law?
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If the elastic constant (k) is determined to be 1.47×10^5 N m^(-1), what does this indicate about the spring's behavior?
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When a spring is compressed from its original length of 30 mm to 24 mm, what is the displacement (s) calculated?
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In the context of simple harmonic motion (SHM), what occurs when the displacement reaches its maximum?
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If a spring is compressed by a force of 882 N resulting in a displacement of -6 mm, what is the elastic constant (k)?
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Which of the following statements accurately describes SHM in terms of acceleration?
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What happens to the velocity of an object in SHM when it passes through the equilibrium position?
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In Hooke's Law, which factor directly influences the restoring force experienced by a spring?
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Which of the following scenarios best exemplifies simple harmonic motion?
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How does the displacement (s) change as a spring is stretched according to Hooke's Law?
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Study Notes
Elasticity and Hooke's Law
- When an object is deformed (stretched, bent, compressed), the restoring force (F) is proportional to the displacement (s) from its equilibrium position.
- The elastic constant (k) relates force and displacement under Hooke's law with SI unit of N m⁻¹.
- Displacement is defined as (s = (l - l_0)), where (l_0) = natural length and (l) = new length.
- Hooke's law formula: (F = k(l - l_0)).
Sample Problem on Elastic Constant
- Calculate elastic constant (k) when a 90 kg mass compresses a spring from 30 mm to 24 mm.
- With (g = 9.8 , m/s^2), use (k = |mg/s|) to find (k = |882 / (-6 \times 10^{-3})| = 1.47 \times 10^5 , N/m).
Simple Harmonic Motion (SHM)
- SHM is periodic motion where acceleration is proportional to and directed towards the equilibrium position; expressed as (a \propto -s).
- Key SHM characteristics:
- At zero displacement, acceleration and forces are zero; velocity is maximal.
- At maximal displacement, acceleration is maximal; velocity is zero.
Parameters of SHM
- Amplitude (A): Maximum distance from equilibrium; measured in meters (m).
- Frequency (f): Number of cycles per time; measured in hertz (Hz); (f = \text{number of oscillations/time taken}).
- Period (T): Time for one complete oscillation; related to frequency (f) by (T = 1/f) or (T = 2\pi/\omega).
Sample Problems on SHM and Pendulum
-
Example with a 500 g mass causing a 25 cm extension on a spring:
- Elastic constant (k) found to be (19.6 , N/m).
-
Determine frequency of oscillation with a 200 g mass:
- Evaluate angular frequency ((\omega = \sqrt{k/m})).
- Period and frequency calculated to be (T = 0.635 , s) and (f \approx 1.6 , Hz).
Pendulum Motion
- The period of a simple pendulum ((T)) depends on its length and gravity:
- (T \propto \sqrt{l}).
- Increasing length increases period; decreasing gravity increases period.
Sample Pendulum Problem
- Calculate period for a pendulum of 50 cm:
- Using (T = 2\pi\sqrt{l/g}) results in (T \approx 1.4 , s).
Experimental Setup for Pendulum
- Apparatus includes pendulum bob, string, and measuring tools.
- Procedure involves measuring bob diameter, setting string length, swinging the pendulum, and timing oscillations.
Data Collection and Analysis
- Calculate periodic time (T) for various lengths, and determine (T^2).
- Plot graph of length ((l)) vs. (T^2) shows linear relationship; slope provides acceleration due to gravity ((g)).
Experimental Precautions
- Ensure small angle (less than 5°) for SHM validity.
- Pendulum should oscillate in one plane with minimal air resistance and constant string length.
- Repeated trials enhance accuracy; use of different masses demonstrates periodic time independence from mass.
Elasticity and Hooke's Law
- When an object is deformed (stretched, bent, compressed), the restoring force (F) is proportional to the displacement (s) from its equilibrium position.
- The elastic constant (k) relates force and displacement under Hooke's law with SI unit of N m⁻¹.
- Displacement is defined as (s = (l - l_0)), where (l_0) = natural length and (l) = new length.
- Hooke's law formula: (F = k(l - l_0)).
Sample Problem on Elastic Constant
- Calculate elastic constant (k) when a 90 kg mass compresses a spring from 30 mm to 24 mm.
- With (g = 9.8 , m/s^2), use (k = |mg/s|) to find (k = |882 / (-6 \times 10^{-3})| = 1.47 \times 10^5 , N/m).
Simple Harmonic Motion (SHM)
- SHM is periodic motion where acceleration is proportional to and directed towards the equilibrium position; expressed as (a \propto -s).
- Key SHM characteristics:
- At zero displacement, acceleration and forces are zero; velocity is maximal.
- At maximal displacement, acceleration is maximal; velocity is zero.
Parameters of SHM
- Amplitude (A): Maximum distance from equilibrium; measured in meters (m).
- Frequency (f): Number of cycles per time; measured in hertz (Hz); (f = \text{number of oscillations/time taken}).
- Period (T): Time for one complete oscillation; related to frequency (f) by (T = 1/f) or (T = 2\pi/\omega).
Sample Problems on SHM and Pendulum
-
Example with a 500 g mass causing a 25 cm extension on a spring:
- Elastic constant (k) found to be (19.6 , N/m).
-
Determine frequency of oscillation with a 200 g mass:
- Evaluate angular frequency ((\omega = \sqrt{k/m})).
- Period and frequency calculated to be (T = 0.635 , s) and (f \approx 1.6 , Hz).
Pendulum Motion
- The period of a simple pendulum ((T)) depends on its length and gravity:
- (T \propto \sqrt{l}).
- Increasing length increases period; decreasing gravity increases period.
Sample Pendulum Problem
- Calculate period for a pendulum of 50 cm:
- Using (T = 2\pi\sqrt{l/g}) results in (T \approx 1.4 , s).
Experimental Setup for Pendulum
- Apparatus includes pendulum bob, string, and measuring tools.
- Procedure involves measuring bob diameter, setting string length, swinging the pendulum, and timing oscillations.
Data Collection and Analysis
- Calculate periodic time (T) for various lengths, and determine (T^2).
- Plot graph of length ((l)) vs. (T^2) shows linear relationship; slope provides acceleration due to gravity ((g)).
Experimental Precautions
- Ensure small angle (less than 5°) for SHM validity.
- Pendulum should oscillate in one plane with minimal air resistance and constant string length.
- Repeated trials enhance accuracy; use of different masses demonstrates periodic time independence from mass.
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Description
Explore the concepts of Hooke's Law and elasticity in this quiz based on physics principles. Understand the relationship between restoring force and displacement, and learn about the elastic constant. Test your knowledge on the mathematical application and significance of elasticity in real-world scenarios.
