The length of a simple pendulum is 0.72 m, the pendulum bob has a mass of 305 grams, and it is released at an angle of 10° to the vertical. a) With what frequency does it vibrate? Assume SHM. (b) What is the pendulum bob's speed when it passes through the lowest point of the swing? (c) What is the total energy stored in this oscillation, assuming no losses?
a) f = [1/(2pi)] sqrt (g/L)
This formula should be in your text or lecture notes.
b) maximum kinetic energy = potential energy when lifted to maximum angle
= M g L (1 - cos 10)
Set that equal to (1/2)M Vmax^2. Cancel the M's and solve for Vmax
c) M g L(1-cos 10)
Thank You drwls :) we did get a formula for frequncy similar to the one you gave me but the one i had didn't work. Once again Thank you for your help :)
a) To find the frequency of the pendulum, we can use the formula:
frequency = 1 / period
The period of a pendulum can be calculated using the formula:
period = 2π * √(length / g)
where g is the acceleration due to gravity (approximately 9.8 m/s²).
Substituting the given values into the formula:
period = 2π * √(0.72 / 9.8)
≈ 2π * 0.303
≈ 1.909 seconds
Therefore, the frequency of the pendulum is:
frequency = 1 / 1.909
≈ 0.524 Hz
b) The speed of the pendulum when it passes through the lowest point of the swing can be calculated using the formula:
speed = √(2 * g * height)
where height is the vertical distance between the highest and lowest points of the swing.
For a simple pendulum, the height is equal to twice the length of the pendulum:
height = 2 * length
= 2 * 0.72
= 1.44 m
Substituting the known values into the formula:
speed = √(2 * 9.8 * 1.44)
= √(28.224)
≈ 5.31 m/s
Therefore, the pendulum bob's speed when passing through the lowest point of the swing is approximately 5.31 m/s.
c) The total energy stored in this oscillation can be calculated using the formula:
energy = 0.5 * mass * speed²
Converting the mass from grams to kilograms:
mass = 305 grams
= 305 / 1000 kg
= 0.305 kg
Substituting the known values into the formula:
energy = 0.5 * 0.305 * (5.31)²
= 0.5 * 0.305 * 28.2161
≈ 2.165 Joules
Therefore, the total energy stored in this oscillation, assuming no losses, is approximately 2.165 Joules.
To answer the given questions, we can utilize the principles of simple harmonic motion (SHM) and the formulas associated with it. Let's break down each question and find the answers step by step.
a) With what frequency does it vibrate?
The frequency of an oscillating pendulum can be calculated using the formula:
Frequency (f) = 1 / Period (T)
To find the period, we can use the formula:
Period (T) = 2π√(L / g)
where L is the length of the pendulum and g is the acceleration due to gravity (approximately 9.8 m/s²).
Using the given information:
Length of the pendulum (L) = 0.72 m
Acceleration due to gravity (g) = 9.8 m/s²
Plugging these values into the formula, we have:
Period (T) = 2π√(0.72 / 9.8)
≈ 2π√0.073
≈ 2π * 0.27
≈ 1.70 seconds (rounded to two decimal places)
Now, we can use the period to find the frequency:
Frequency (f) = 1 / T
= 1 / 1.70
≈ 0.59 Hz (rounded to two decimal places)
Therefore, the frequency of vibration is approximately 0.59 Hz.
b) What is the pendulum bob's speed when it passes through the lowest point of the swing?
At the lowest point of the swing, the potential energy is minimum and all the energy is converted to kinetic energy.
The speed of the pendulum bob at the lowest point can be calculated using the formula:
Velocity (v) = √(2gh)
where h is the height (distance) above the lowest point of the swing.
In this case, the height (h) is equal to the length of the pendulum (L).
Using the given information:
Length of the pendulum (L) = 0.72 m
Acceleration due to gravity (g) = 9.8 m/s²
Plugging these values into the formula, we have:
Velocity (v) = √(2 * 9.8 * 0.72)
≈ √(13.824)
≈ 3.72 m/s (rounded to two decimal places)
Therefore, the pendulum bob's speed when it passes through the lowest point of the swing is approximately 3.72 m/s.
c) What is the total energy stored in this oscillation, assuming no losses?
The total energy (E) in the pendulum's oscillation is the sum of its potential energy and kinetic energy.
The potential energy (PE) of a pendulum can be calculated using the formula:
Potential Energy (PE) = mgh
where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the height (distance) above the lowest point of the swing.
Using the given information:
Mass of the pendulum bob (m) = 305 grams (convert to kilograms: 0.305 kg)
Acceleration due to gravity (g) = 9.8 m/s²
Height above the lowest point (h) = length of the pendulum (L) = 0.72 m
Plugging these values into the formula, we have:
Potential Energy (PE) = 0.305 * 9.8 * 0.72
≈ 2.11 J (rounded to two decimal places)
Since the sum of potential energy (PE) and kinetic energy (KE) gives us the total energy (E), and at the lowest point of the swing, all energy is converted to kinetic energy, the total energy will be equal to the kinetic energy at the lowest point.
Hence, the total energy stored in this oscillation is approximately 2.11 J.