Energy in Pendulums: Videos & Practice Problems

In pendulum motion, energy oscillates between kinetic energy and gravitational potential energy. At maximum height, gravitational potential energy is highest, while kinetic energy is zero. Conversely, at the lowest point, kinetic energy peaks and gravitational potential energy is zero. The height can be calculated using the equation: h_max = l(1−cosθ). The conservation of energy principle applies, allowing the derivation of maximum speed as v_max = √2g(l(1−cosθ)).

Energy in Pendulums

In the study of pendulums, understanding the interplay between kinetic energy and gravitational potential energy is crucial. As a pendulum swings, it oscillates between two energy states: at its highest point, where it has maximum gravitational potential energy and zero kinetic energy, and at its lowest point, where it has maximum kinetic energy and zero gravitational potential energy. The gravitational potential energy (U) can be expressed as:

U = mgh

where m is the mass, g is the acceleration due to gravity, and h is the height above a defined zero point, typically the lowest point of the swing.

When the pendulum is pulled back to an angle θ and released, it swings back and forth, reaching a maximum height at the angle θ. The height h at any angle can be determined using the pendulum equation:

h = l(1 - \cos(θ))

where l is the length of the pendulum. This equation derives from the geometry of the pendulum's motion, utilizing the cosine function to find the vertical height relative to the lowest point.

The total mechanical energy (E) of the pendulum is conserved and can be expressed as:

E = U + K = mgh + \frac{1}{2}mv^2

At the maximum height, the energy is entirely potential, while at the lowest point, it is entirely kinetic. Thus, we can set up the energy conservation equation:

mg(l(1 - \cos(θ))) = \frac{1}{2}mv_{max}^2

By simplifying this equation, we can derive the maximum speed (v_max) of the pendulum as it passes through the lowest point:

v_{max} = \sqrt{2gl(1 - \cos(θ))}

This formula allows us to calculate the maximum speed of the pendulum based on its length and the angle from which it was released. Understanding these principles is essential for solving problems related to pendulum motion and energy conservation.

Example

In this exercise, we explore the dynamics of a pendulum to determine its maximum height based on given parameters. We start with a pendulum of length \( L = 2 \) meters and a mass \( m = 0.4 \) kg, which is at a 5-degree angle from the vertical while moving at a speed of \( v = 1.5 \) m/s. The goal is to find the maximum height \( h_{\text{max}} \) that the pendulum reaches during its swing.

To solve this, we apply the principle of conservation of mechanical energy, which states that the total mechanical energy in a closed system remains constant. The relevant equation for energy conservation in this context is:

\[ mgh_{\text{max}} = mgh_p + \frac{1}{2} mv^2 \]

Here, \( h_p \) is the height of the pendulum at the specific point where it makes a 5-degree angle. To find \( h_p \), we use the formula:

\[ h_p = L(1 - \cos(\theta_p)) \]

Before substituting values, we need to convert the angle from degrees to radians:

\[ \theta_p = 5 \times \frac{\pi}{180} \approx 0.087 \text{ radians} \]

Now, substituting the values into the height equation:

\[ h_p = 2(1 - \cos(0.087)) \approx 2(1 - 0.9962) \approx 0.0076 \text{ meters} \]

With \( h_p \) calculated, we can now substitute back into the energy conservation equation. Rearranging gives us:

\[ h_{\text{max}} = h_p + \frac{v^2}{2g} \]

Substituting the known values, where \( g \approx 9.8 \, \text{m/s}^2 \):

\[ h_{\text{max}} = 0.0076 + \frac{(1.5)^2}{2 \times 9.8} \]

Calculating the kinetic energy term:

\[ \frac{(1.5)^2}{2 \times 9.8} = \frac{2.25}{19.6} \approx 0.1153 \]

Now, adding this to \( h_p \):

\[ h_{\text{max}} = 0.0076 + 0.1153 \approx 0.1229 \text{ meters} \]

Thus, the maximum height \( h_{\text{max}} \) that the pendulum reaches is approximately \( 0.12 \) meters. This exercise illustrates the application of energy conservation principles in analyzing pendulum motion, emphasizing the relationship between potential and kinetic energy at different points in the swing.

A mass swinging at the end of a pendulum has a speed of 1.32m/s at the bottom of its swing. At the top of its swing, it makes a 9° with the vertical. What is the length of the pendulum?

• A. 6.83 m
• B. 7.22 m
• C. 1.77 m
• D. 5.18 m

Answer: (B)

Here’s what students ask on this topic:

What is the energy conservation formula for a pendulum?

The energy conservation formula for a pendulum states that the total mechanical energy (sum of kinetic and potential energy) remains constant throughout its motion. At any point in the pendulum's swing, the total energy can be expressed as:

$E_{\mathrm{total}}=U_g+K_e$

Where $U_g$ is the gravitational potential energy and $K_e$ is the kinetic energy. At the highest point, the energy is all potential:

$E_{\mathrm{total}}=U_g=mgh$

At the lowest point, the energy is all kinetic:

$E_{\mathrm{total}}=K_e=\frac{1}{2}m{v}^2$

How do you calculate the maximum height of a pendulum?

The maximum height $h_{\max }$ of a pendulum can be calculated using the equation:

$h_{\max }=l(1-cos\theta )$

Here, $l$ is the length of the pendulum and $\theta$ is the maximum angle from the vertical. This equation derives from the geometry of the pendulum's swing, where the height is the difference between the pendulum's length and the vertical component of the length when displaced by angle $\theta$.

What is the maximum speed of a pendulum?

The maximum speed $v_{\max }$ of a pendulum can be derived using energy conservation principles. The formula is:

$v_{\max }=\sqrt{2gl(1-cos\theta )}$

Here, $g$ is the acceleration due to gravity, $l$ is the length of the pendulum, and $\theta$ is the maximum angle from the vertical. This formula shows that the maximum speed occurs at the lowest point of the swing, where all the potential energy has been converted to kinetic energy.

How does the length of a pendulum affect its energy?

The length $l$ of a pendulum significantly affects its energy. The gravitational potential energy at the maximum height is given by:

$U_g=mg{h}_{\max }$

Since $h_{\max }=l(1-cos\theta )$, the potential energy depends directly on the length of the pendulum. A longer pendulum will have a higher maximum height for the same angle $\theta$, resulting in greater potential energy. Consequently, the kinetic energy at the lowest point, which is equal to the initial potential energy, will also be higher for a longer pendulum.

What is the relationship between kinetic and potential energy in a pendulum?

In a pendulum, kinetic energy (KE) and potential energy (PE) oscillate between each other. At the highest point of the swing, the pendulum has maximum potential energy and zero kinetic energy:

$U_g=mg{h}_{\max }$

At the lowest point, the pendulum has maximum kinetic energy and zero potential energy:

$K_e=\frac{1}{2}m{v}^2$

Throughout the swing, the total mechanical energy (sum of KE and PE) remains constant, illustrating the conservation of energy principle. As the pendulum moves, energy continuously converts between kinetic and potential forms.