A simple pendulum is made from a ping-pong ball with a mass of 10 grams, attached to a 60-cm length of thread with negligible mass. The force of air resistance on the ball is Fair =−b𝑥̇, in which = 0.016 kg s-1 [gravitational acceleration, g = 9.81 ms-2]

Question

A simple pendulum is made from a ping-pong ball with a mass of 10 grams, attached to a 60-cm length of thread with negligible mass. The force of air resistance on the ball is Fair =−b𝑥̇, in which = 0.016 kg s-1 [gravitational acceleration, g = 9.81 ms-2]

(a) Show that the pendulum is underdamped. Find the angular frequency, ω and the period, T of oscillation, (b) How long does it take for the amplitude of the pendulum’s swing to decrease by a factor of 1000? (c) If a pendulum made with the same ping-pong ball were to be critically damped by air resistance, what would its length have to be?
(d) An external harmonic force, F(t) = Fo cos(ωt), is applied to the ping-pong pendulum of to keep it swinging despite the air resistance. What driving angular frequency and force amplitude, Fo are required if the pendulum is to have its natural period (T0= 2π/ω0) and angular amplitude of 10o in the steady state?

Answer 1

(a) Well, the pendulum is underdamped because it experiences air resistance. Air resistance acts against the motion, which causes the pendulum to slow down, resulting in a gradual decrease in amplitude over time.

To find the angular frequency, ω, we can start by considering the forces acting on the pendulum. The net force is given by:

Fnet = Tension - Fair - mgθ

where Tension is the tension in the thread, Fair is the force of air resistance, mg is the weight of the ball, and θ is the angle of displacement.

Since we are dealing with small angles, we can approximate sin(θ) ≈ θ. Thus, the equation can be simplified to:

Fnet = Tension - b𝑥̇ - mgθ

Now, using Newton's second law, Fnet = ma, where a is the acceleration and m is the mass of the ball, we can set up the differential equation:

ma = Tension - b𝑥̇ - mgθ

Since the only force acting in the horizontal direction is the force of air resistance, we can rewrite the equation as:

ma = -b𝑥̇

Using the relation ẍ = d²x/dt² and substituting in the values, we get:

mẍ = -b𝑥̇

Dividing the equation by m, we have:

ẍ + (b/m)𝑥̇ = 0

Comparing this equation to the standard form of the underdamped oscillator, ẍ + 2ζωnẋ + ωn²x = 0, we can see that (b/m) = 2ζωn, where ζ is the damping ratio and ωn is the natural frequency.

Since we are dealing with a pendulum, the natural frequency is given by ωn = √(g/L), where L is the length of the thread.

Hence, (b/m) = 2ζ√(g/L)

(b) The time taken for the amplitude of the pendulum's swing to decrease by a factor of 1000, also known as the "damping time," can be found using the formula:

τ = 1 / (ζωn)

Given that ζ = 1/1000, we can substitute the value of ωn = √(g/L) to find τ.

(c) In the case of critical damping, the damping ratio is ζ = 1. To find the length L required for critical damping, we can substitute ζ = 1 into the earlier equation:

(b/m) = 2ζ√(g/L)

Simplifying and solving for L, we can find the length needed for critical damping.

(d) To keep the pendulum swinging despite air resistance, an external harmonic force is applied. The driving angular frequency ωd and force amplitude Fo are required such that the pendulum has its natural period T0 = 2π/ω0 and an angular amplitude of 10o in the steady state.

We can start by considering the equation for the net force:

Fnet = Tension - Fair - mgθ

Since we want to balance out the damping force and maintain the pendulum's motion, we can have:

Fnet = -b𝑥̇ + Fo cos(ωd t) - mgθ

Substituting in the approximations sin(θ) ≈ θ and Tension = mgθ, we get:

Fnet = -b𝑥̇ + Fo cos(ωd t) - mgθ ≈ -b𝑥̇ + Fo cos(ωd t) - Tension

Using Newton's second law, Fnet = ma, where a is the acceleration, we can write the differential equation:

ma = -b𝑥̇ + Fo cos(ωd t) - Tension

Now, substituting ẍ = d²x/dt² and solving the equation, we can find the values of ωd and Fo that satisfy the given conditions.

Answer 2

(a) To determine if the pendulum is underdamped, we need to examine the damping coefficient, b, and the angular frequency, ω, of the pendulum's motion.

The equation of motion for the pendulum is given by:

m𝑥̈ + b𝑥̇ + mg𝑥 = 0

where m is the mass of the pendulum, 𝑥̈ is the acceleration of the pendulum, 𝑥̇ is the velocity of the pendulum, and 𝑥 is the displacement of the pendulum.

In this case, the mass of the ping-pong ball is 10 grams, which is equivalent to 0.01 kg.

The force of air resistance is given by Fair = -b𝑥̇.

We can rewrite this force as m𝑥̈ + b𝑥̇ = -mg𝑥.

Since g = 9.81 m/s^2, we can substitute the values into the equation:

0.01 𝑥̈ + b𝑥̇ = -0.01 * 9.81 𝑥

Divide through by 0.01:

𝑥̈ + b𝑥̇ = -0.981 𝑥

The characteristic equation for the damped harmonic oscillator is:

𝑟^2 + 𝛽𝑟 + 𝜔^2 = 0

where 𝑟 is the damping ratio, 𝛽 is the damping coefficient, and 𝜔 is the angular frequency.

Comparing this with our equation, we have:

1𝑟^2 + b𝑟 + 𝜔^2 = 0

This is a second-order differential equation, which implies that the pendulum is underdamped.

To find ω, we can compare coefficients:

𝜔^2 = 0.981
∴ 𝜔 = √0.981 ≈ 0.990 rad/s

The period of a simple pendulum is given by T = 2π/ω:

T = 2π/0.990
≈ 2π * 1.010
≈ 6.343 s

(b) The amplitude of the pendulum's swing decreases by a factor of 1000 after a time equal to three times the damping time constant.

The damping time constant is given by τ = 1/b.

Substituting the given damping coefficient b = 0.016 kg/s into the equation:

τ = 1/0.016
= 62.5 s

Therefore, the time taken for the amplitude to decrease by a factor of 1000 is:

3 * τ
= 3 * 62.5
= 187.5 s

(c) To make the pendulum critically damped, the damping coefficient, b, would need to be equal to 2√m𝑔.

Substituting the values of m = 0.01 kg and g = 9.81 m/s^2:

b = 2√(0.01 * 9.81)
≈ 0.62 kg/s

To find the corresponding length, we can use the relation between the damping coefficient and the length of the pendulum.

For a critically damped pendulum:

b = 2√(𝑔/𝐿)

Rearranging the equation:

𝐿 = 2√(𝑔/b)
≈ 2√(9.81/0.62)
≈ 6.29 m

Therefore, the length of the pendulum would need to be approximately 6.29 meters to be critically damped.

(d) In the steady-state, the external harmonic force balances the damping force.

The equation of motion for the damped harmonic oscillator with an external force is:

m𝑥̈ + b𝑥̇ + k𝑥 = Fo cos(ωt)

where k is the spring constant and Fo is the amplitude of the external driving force.

In our case, the force of air resistance is given by Fair = -b𝑥̇, so the damping coefficient, b, replaces k in the equation.

Comparing coefficients, we have:

b = Fo

To have the natural period (T0 = 2π/ω0), we can equate the angular frequency with ω0:

ω = ω0

Therefore, we have:

Fo = ω m

To have an angular amplitude of 10°, we can equate the angular frequency with ω:

10° = ω

Substituting into the equation Fo = ω m:

Fo = (10°) * (0.01 kg) ≈ 0.0017 kg m/s^2

Therefore, the driving angular frequency required is approximately 10°, and the force amplitude required is approximately 0.0017 kg m/s^2.

Answer 3

To answer these questions, we need to understand the behavior of a simple pendulum under various conditions. We'll go through each question one by one.

(a) To determine whether the pendulum is underdamped, we need to analyze the behavior of the damping force. The damping force is given by Fair = -b𝑥̇, where b = 0.016 kg s^-1 represents the damping coefficient and 𝑥̇ represents the velocity of the pendulum bob.

For a simple pendulum, the equation of motion is given by:

m𝑎 = -mg sin(𝜃) - b𝑥̇

Here, m represents the mass of the pendulum bob, g represents the gravitational acceleration, and 𝜃 represents the angular displacement. We can substitute a = 𝑥̈, the acceleration of the pendulum bob.

m𝑥̈ = -mg sin(𝜃) - b𝑥̇

For small angles, sin(𝜃) ≈ 𝜃. Thus, we can rewrite the equation of motion as:

m𝑥̈ = -mg𝜃 - b𝑥̇

Now, let's substitute m = 10 grams = 0.01 kg and g = 9.81 m/s^2 into the equation:

0.01𝑥̈ = -0.01 * 9.81 * 𝜃 - 0.016𝑥̇

Dividing the entire equation by 0.01, we get:

𝑥̈ = -9.81𝜃 - 1.6𝑥̇

Comparing this equation with the general form of the equation for underdamped harmonic motion:

𝑥̈ + 2𝜁ω𝑥̇ + ω^2𝑥 = 0

We can see that ω^2 = 9.81, which gives us the angular frequency ω = sqrt(9.81). This represents the natural frequency of the pendulum's oscillation.

(b) The amplitude of a simple pendulum's swing decreases exponentially over time due to damping. The damping ratio (𝜁) can be calculated using the formula:

𝜁 = b / (2mω)

Substituting the given values, we have 𝜁 = 0.016 / (2 * 0.01 * sqrt(9.81)).

The time it takes for the amplitude to decrease by a factor of 1000 is given by:

T_decay = (ln(1000) / 𝜁) / ω

Substituting the values, we can calculate the time.

(c) To make the pendulum critically damped, the damping ratio (𝜁) should be equal to 1. The length (L) of the pendulum can be calculated using the formula:

L = (g * 𝑇𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙^2) / (4𝜋^2)

Substituting the values, we find the required length.

(d) To keep the pendulum swinging despite the air resistance, an external harmonic force is applied. The driving angular frequency (ω_d) and force amplitude (Fo) can be determined by matching the natural period (T_0) and the angular amplitude (𝜃_0).

The natural period is given by T_0 = 2𝜋/ω_0, where ω_0 is the natural angular frequency.

Substituting the given values, we can determine ω_0.

In the steady state, the driving force should match the natural frequency of the pendulum, so ω_d should be equal to ω_0. The force amplitude Fo can be determined by comparing the amplitudes (A) of the driving force and the pendulum's steady-state response:

Fo/m = A_d/A

Where A_d is the amplitude of the driving force and A is the amplitude of the pendulum's steady-state response.

Substituting the given values, we can calculate Fo and ω_d.

By following these steps and performing the necessary calculations, you should be able to find the answers to all the questions related to the simple pendulum.

A simple pendulum is made from a ping-pong ball with a mass of 10 grams, attached to a 60-cm length of thread with negligible mass. The force of air resistance on the ball is Fair =−b𝑥̇, in which = 0.016 kg s-1 [gravitational acceleration, g = 9.81 ms-2]

(a) Well, the pendulum is underdamped because it experiences air resistance. Air resistance acts against the motion, which causes the pendulum to slow down, resulting in a gradual decrease in amplitude over time.

(a) To determine if the pendulum is underdamped, we need to examine the damping coefficient, b, and the angular frequency, ω, of the pendulum's motion.

To answer these questions, we need to understand the behavior of a simple pendulum under various conditions. We'll go through each question one by one.

babi