Simple Harmonic Motion of Pendulums: Videos & Practice Problems

Simple harmonic motion is exemplified by pendulums, which oscillate between maximum angles (amplitudes) when displaced. The restoring force is derived from gravity, expressed as F = -mgsin(θ), simplifying to F = -mgθ for small angles. The period of oscillation is given by T = 2πlg. Understanding these relationships is crucial for analyzing pendulum dynamics.

Simple Harmonic Motion of Pendulums

Simple harmonic motion can be observed in both mass-spring systems and simple pendulums, with each exhibiting similar characteristics but differing in their specific parameters. In a mass-spring system, when a mass is displaced from its equilibrium position and released, it oscillates back and forth between maximum displacements, known as amplitudes. For a simple pendulum, the process is analogous; however, instead of a spring constant, we consider the length of the pendulum, denoted as \( l \), and the angle of displacement, \( \theta \).

When a pendulum is pulled to an angle \( \theta \) and released, it oscillates between two maximum angles, which are the positive and negative amplitudes. The restoring force acting on the pendulum is derived from the gravitational force acting on the mass, which can be expressed as \( F = -mg \sin(\theta) \). Here, \( m \) represents the mass of the pendulum bob, and \( g \) is the acceleration due to gravity. The acceleration of the pendulum can thus be described by the equation \( a = -g \sin(\theta) \).

For small angles, the sine function can be approximated by the angle itself when measured in radians, allowing us to simplify the restoring force to \( F = -mg\theta \) and the acceleration to \( a = -g\theta \). This approximation is crucial for maintaining the linear relationship required for simple harmonic motion.

The angular frequency \( \omega \) for a simple pendulum is given by the formula \( \omega = \sqrt{\frac{g}{l}} \), contrasting with the mass-spring system where \( \omega = \sqrt{\frac{k}{m}} \). This distinction can be remembered by noting the alphabetical order of the variables: \( k \) comes before \( m \) and \( g \) comes before \( l \).

To find the period of oscillation \( T \) for a simple pendulum, the relationship can be expressed as \( T = 2\pi \sqrt{\frac{l}{g}} \). For example, if a pendulum has a length of \( 0.25 \, \text{m} \) and a mass of \( 4 \, \text{kg} \), the period can be calculated using the aforementioned formula, resulting in a period of approximately \( 1.00 \, \text{s} \).

Additionally, the time taken for the pendulum to reach its maximum speed occurs at a quarter of the period. Since the full period is \( 1.00 \, \text{s} \), the time to reach maximum speed is \( 0.25 \, \text{s} \). This understanding of the dynamics of a simple pendulum is essential for solving related problems in physics.

Example

In this example, we explore the dynamics of a simple pendulum on an unfamiliar planet. The pendulum has a length of \( l = 3 \) meters and a mass of \( m = 4 \) kilograms. It is released from an angle of \( \theta_{\text{max}} = 10^\circ \) from the vertical, which is important to note as we will need to convert this angle into radians for calculations. The period of oscillation, which is the time taken for one complete cycle, is given as \( T = 2 \) seconds.

To determine the acceleration due to gravity (\( g \)) on this planet, we can utilize the relationship between the period of a pendulum and gravitational acceleration. The formula that relates these variables is:

\[ T = 2\pi \sqrt{\frac{l}{g}} \]

Rearranging this equation allows us to express \( g \) in terms of the period and length:

\[ g = \frac{4\pi^2 l}{T^2} \]

Substituting the known values into the equation, we have:

\[ g = \frac{4\pi^2 \cdot 3}{(2)^2} \]

Calculating this gives:

\[ g = \frac{4\pi^2 \cdot 3}{4} = 3\pi^2 \]

Now, evaluating \( 3\pi^2 \) yields approximately \( 29.6 \, \text{m/s}^2 \). This value is significantly higher than Earth's gravitational acceleration of \( 9.8 \, \text{m/s}^2 \), indicating that the gravitational force on this planet is much stronger.

In summary, by applying the principles of pendulum motion and the relationship between period and gravitational acceleration, we successfully calculated the gravity on the new planet, demonstrating the importance of understanding these fundamental concepts in physics.

Pendulum Equations

In analyzing the motion of pendulums and mass-spring systems, we observe that both systems exhibit harmonic motion, characterized by specific relationships between displacement, velocity, and acceleration. For a pendulum of length \( l \), when displaced to an angle \( \theta \), the horizontal displacement \( x \) can be expressed using trigonometric relationships. Specifically, \( x \) is related to the angle by the equation:

\( x = l \sin(\theta) \)

For small angles, \( \sin(\theta) \) approximates \( \theta \) (in radians), allowing us to simplify this to:

\( x \approx l \theta \

This means that the maximum displacement, or amplitude \( x_{\text{max}} \), occurs at the maximum angle \( \theta_{\text{max}} \), leading to the relationship:

\( x_{\text{max}} = l \theta_{\text{max}} \

In both systems, the maximum velocity \( v_{\text{max}} \) occurs at the equilibrium position, while the maximum acceleration \( a_{\text{max}} \) occurs at the endpoints of the swing. The equations governing these relationships are similar for both systems. For pendulums, the maximum velocity can be expressed as:

\( v_{\text{max}} = a \omega \

where \( a \) is the amplitude and \( \omega \) is the angular frequency. Substituting \( a \) with \( l \theta_{\text{max}} \), we have:

\( v_{\text{max}} = l \theta_{\text{max}} \omega \

Similarly, the maximum acceleration can be expressed as:

\( a_{\text{max}} = a \omega^2 = l \theta_{\text{max}} \omega^2 \

For pendulums, the angular frequency \( \omega \) is determined by the formula:

\( \omega = \sqrt{\frac{g}{l}} \

where \( g \) is the acceleration due to gravity. This contrasts with mass-spring systems, where \( \omega \) is given by:

\( \omega = \sqrt{\frac{k}{m}} \

where \( k \) is the spring constant and \( m \) is the mass. When solving for the maximum angle \( \theta_{\text{max}} \) at a given maximum velocity, we can rearrange the equation for maximum velocity:

\( \theta_{\text{max}} = \frac{v_{\text{max}}}{l \omega} \

To find \( \omega \), we substitute \( \omega = \sqrt{\frac{g}{l}} \) into the equation. For example, if we have a pendulum with a mass of 0.5 kg and a length of 0.4 m, and it passes through the lowest point with a speed of 0.25 m/s, we can calculate \( \omega \) as follows:

\( \omega = \sqrt{\frac{9.8}{0.4}} \approx 4.95 \, \text{rads/s} \

Substituting this back into the equation for \( \theta_{\text{max}} \), we find:

\( \theta_{\text{max}} = \frac{0.25}{0.4 \times 4.95} \approx 0.126 \, \text{rads} \

Finally, converting radians to degrees involves multiplying by \( \frac{180}{\pi} \), yielding:

\( \theta_{\text{max}} \approx 7.2^\circ \

This analysis illustrates the fundamental principles governing the motion of pendulums and mass-spring systems, highlighting their similarities and the mathematical relationships that define their behavior.

Example

In this pendulum problem, we analyze a mass of 100 grams attached to a 1-meter long string, initially displaced to 7 degrees from its equilibrium position. The goal is to determine the time it takes for the pendulum to swing to a position where it reaches -4 degrees on the opposite side.

To begin, we establish the parameters of the pendulum: the mass (m) is 0.1 kg (converted from grams), the length (L) is 1 meter, the maximum angle (θ_max) is 7 degrees, and the final angle (θ_final) is -4 degrees. It is crucial to convert these angles from degrees to radians for calculations, using the conversion factor π/180. Thus, θ_max becomes approximately 0.12 radians, and θ_final becomes approximately -0.07 radians.

We utilize the equation for angular displacement in simple harmonic motion, which is given by:

θ(t) = θ_max * cos(ωt)

Here, θ(t) represents the angle at time t, and ω is the angular frequency. Rearranging the equation to solve for time (t), we have:

cos(ωt) = θ(t) / θ_max

Substituting the known values:

cos(ωt) = -0.07 / 0.12

Calculating this gives us cos(ωt) ≈ -0.58. To isolate ωt, we apply the inverse cosine function:

ωt = cos^-1(-0.58)

This results in ωt ≈ 2.19 radians.

Next, we need to determine the angular frequency (ω). For a simple pendulum, ω can be calculated using the formula:

ω = √(g / L)

Where g is the acceleration due to gravity (approximately 9.8 m/s²) and L is the length of the pendulum (1 meter). Thus:

ω = √(9.8 / 1) ≈ 3.13 radians/second.

Now, substituting ω back into the equation for t:

2.19 = 3.13t

Solving for t gives:

t = 2.19 / 3.13 ≈ 0.7 seconds.

This result indicates that it takes approximately 0.7 seconds for the pendulum to swing from 7 degrees to -4 degrees on the opposite side. Understanding the principles of pendulum motion, including angular displacement and frequency, is essential for solving similar problems in physics.

Here’s what students ask on this topic:

What is the formula for the period of a simple pendulum?

The period of a simple pendulum, which is the time it takes for one complete oscillation, is given by the formula:

$T=2\pi \sqrt{\frac{l}{g}}$

Here, $l$ is the length of the pendulum and $g$ is the acceleration due to gravity (approximately 9.8 m/s² on Earth). This formula assumes that the amplitude of the oscillation is small, so the approximation $\sin \theta \approx \theta$ holds true.

How does the length of a pendulum affect its period?

The period of a pendulum is directly related to the square root of its length. According to the formula:

$T=2\pi \sqrt{\frac{l}{g}}$

As the length $l$ increases, the period $T$ also increases. Specifically, if you double the length of the pendulum, the period increases by a factor of $\sqrt{2}$. This relationship shows that longer pendulums swing more slowly than shorter ones.

What is the restoring force in a simple pendulum?

The restoring force in a simple pendulum is the component of the gravitational force that acts to bring the pendulum back to its equilibrium position. For small angles, this force can be approximated as:

$F=-\mathrm{mg}\theta$

Here, $m$ is the mass of the pendulum bob, $g$ is the acceleration due to gravity, and $\theta$ is the angular displacement in radians. This approximation holds true for small angles where $\sin \theta \approx \theta$.

How do you convert the angle of displacement from degrees to radians in a pendulum problem?

To convert the angle of displacement from degrees to radians, you can use the conversion factor:

$\theta =\frac{\pi \theta °}{180}$

Here, $\theta$ is the angle in radians and $\theta °$ is the angle in degrees. For example, to convert 3.5 degrees to radians:

$\theta =\frac{3.5\pi }{180}\approx 0.061$

What is the relationship between angular frequency and the length of a pendulum?

The angular frequency $\omega$ of a simple pendulum is related to the length of the pendulum by the formula:

$\omega =\sqrt{\frac{g}{l}}$

Here, $g$ is the acceleration due to gravity and $l$ is the length of the pendulum. This shows that the angular frequency is inversely proportional to the square root of the length. As the length increases, the angular frequency decreases, meaning the pendulum swings more slowly.