A 0.60 kg mass is attached to a 0.6 m string and displaced at an angle of 15 degrees before it is released. 1)What is the potential energy of the pendulum? 2) What is the angular frequency of the pendulum? What is the height of its displacement? 4) What is the velocity of the pendulum at the lowest point in its swing?
To answer these questions, we need to apply the concepts of potential energy, angular frequency, height, and velocity in the context of a pendulum. Let's address each question step by step:
1) What is the potential energy of the pendulum?
The potential energy of an object in the context of a pendulum is given by the formula:
Potential Energy = m * g * h
where m is the mass of the object, g is the acceleration due to gravity, and h is the height of displacement.
In this case, the mass (m) is given as 0.60 kg. The acceleration due to gravity (g) can be approximated as 9.8 m/s². The height (h) can be calculated from the displacement angle of 15 degrees and the length of the string, using trigonometry.
h = length of string * sin(displacement angle)
Given that the length of the string is 0.6 m, we can compute h as follows:
h = 0.6 m * sin(15°)
Finally, substitute the values into the potential energy formula:
Potential Energy = 0.60 kg * 9.8 m/s² * (0.6 m * sin(15°))
2) What is the angular frequency of the pendulum?
The angular frequency (ω) of a pendulum can be calculated using the formula:
Angular Frequency = √(g / L)
where g is the acceleration due to gravity and L is the length of the string.
In this case, g is again approximately 9.8 m/s², and the length of the string (L) is given as 0.6 m.
Angular Frequency = √(9.8 m/s² / 0.6 m)
3) What is the height of its displacement?
The height (h) can be calculated using the displacement angle and the length of the string. We have already discussed the formula for calculating h in question 1.
h = length of string * sin(displacement angle)
In this case, the displacement angle is given as 15 degrees, and the length of the string is 0.6 m. Therefore,
h = 0.6 m * sin(15°)
4) What is the velocity of the pendulum at the lowest point in its swing?
To calculate the velocity (v) of the pendulum at the lowest point in its swing, we can use the conservation of energy. Since the potential energy at the highest point is converted entirely into kinetic energy at the lowest point, we can set the potential energy equal to the kinetic energy equation.
Potential Energy = Kinetic Energy
Using the formula for potential energy from question 1 and the formula for kinetic energy (Kinetic Energy = (1/2) * m * v^2), we can write:
m * g * h = (1/2) * m * v^2
Simplify and solve for v:
v = √(2 * g * h)
Now, substitute the values into the formula:
v = √(2 * 9.8 m/s² * h)
To solve these questions, we need to use the formulas related to the potential energy, angular frequency, height, and velocity of a pendulum.
1) The potential energy of the pendulum can be calculated using the formula:
Potential energy = mass × gravitational acceleration × height
In this case, the mass (m) is given as 0.60 kg, the gravitational acceleration (g) is approximately 9.8 m/s^2 (standard value near the Earth's surface), and the height (h) can be determined using the displacement (d) and the angle (θ) as follows:
height = displacement × sin(angle)
Given the displacement (d) as 0.6 m and the angle (θ) as 15 degrees, we have:
height = 0.6 m × sin(15°) ≈ 0.154 m
Now we can calculate the potential energy:
Potential energy = 0.60 kg × 9.8 m/s^2 × 0.154 m ≈ 0.9 J
Therefore, the potential energy of the pendulum is approximately 0.9 Joules.
2) The angular frequency (ω) relates to the period (T) of the pendulum according to the formula:
Angular frequency = 2π / Period
The period (T) of a simple pendulum can be calculated using the formula:
Period = 2π × √(length / gravitational acceleration)
Given the length of the string (L) as 0.6 m and the gravitational acceleration (g) as 9.8 m/s^2, we have:
Period = 2π × √(0.6 / 9.8) ≈ 1.239 s
Now we can calculate the angular frequency:
Angular frequency = 2π / 1.239 ≈ 5.068 rad/s
Therefore, the angular frequency of the pendulum is approximately 5.068 radians per second.
3) The height of the pendulum's displacement can be calculated using the same formula as before:
height = displacement × sin(angle)
Given the displacement (d) as 0.6 m and the angle (θ) as 15 degrees, we have:
height = 0.6 m × sin(15°) ≈ 0.154 m
Therefore, the height of the pendulum's displacement is approximately 0.154 meters.
4) The velocity of the pendulum at the lowest point in its swing can be calculated using the formula:
Velocity = Angular frequency × amplitude
The amplitude (A) of the pendulum is directly related to its displacement, so we can use the same value for the displacement (d) as before:
amplitude = displacement = 0.6 m
Now we can calculate the velocity:
Velocity = 5.068 rad/s × 0.6 m ≈ 3.041 m/s
Therefore, the velocity of the pendulum at the lowest point in its swing is approximately 3.041 meters per second.