A 25.0-g mass is tied at the end of a 1.50-m long string and is whirled in a horizontal circle. If the string makes an angle of 30.0o with the vertical, what is the frequency of revolution of the mass?
To find the frequency of revolution of the mass, we can use the formula:
ω = √(g/L)
Where:
ω is the angular velocity (in radians per second),
g is the acceleration due to gravity (approximately 9.81 m/s²),
L is the length of the string.
First, let's convert the angle from degrees to radians:
θ = 30.0° * π/180
θ = 0.5236 radians
Now, we can calculate the length of the vertical component of the string:
L_vertical = L * sin(θ)
L_vertical = 1.50 m * sin(0.5236)
L_vertical = 0.795 m
Since the mass is whirled in a horizontal circle, the tension in the string provides the centripetal force. The horizontal component of the string length will not affect the circular motion.
Now, we can find the effective length of the string:
L_eff = L - L_vertical
L_eff = 1.50 m - 0.795 m
L_eff = 0.705 m
Finally, we can calculate the frequency of revolution:
ω = √(g/L_eff)
ω = √(9.81 m/s² / 0.705 m)
ω ≈ 4.643 rad/s
The frequency of revolution, f, is related to the angular velocity by the equation:
f = ω / (2π)
So, we can calculate the frequency:
f ≈ 4.643 rad/s / (2π)
f ≈ 0.739 Hz
Therefore, the frequency of revolution of the mass is approximately 0.739 Hz.
To find the frequency of revolution of the mass, we first need to determine the speed at which the mass is moving in its circular path. We can use the information provided to solve this problem using the following steps:
Step 1: Draw a diagram to visualize the problem.
- Represent the vertical string as a line extending from the center of the circle or pivot point.
- The string makes an angle of 30.0 degrees with the vertical, which means it makes an angle of 60.0 degrees with the horizontal direction.
- Attach the 25.0-g mass at the end of the string.
Step 2: Decompose the gravitational force acting on the mass into its horizontal and vertical components.
- The vertical component of the gravitational force balances the tension in the string, while the horizontal component provides the centripetal force to keep the mass moving in a circle.
Step 3: Calculate the horizontal component of the gravitational force.
- The horizontal component of the gravitational force is given by F_h = mg * sin(angle), where m is the mass and g is the acceleration due to gravity.
- In this case, F_h = (25.0 g) * (9.8 m/s^2) * sin(60.0°).
Step 4: Calculate the centripetal force.
- The centripetal force is given by F_c = m * (v^2 / r), where m is the mass, v is the velocity, and r is the radius of the circular path.
- In this case, F_c = (25.0 g) * (v^2 / 1.50 m), where v is the speed at which the mass is moving.
Step 5: Equate the centripetal force to the horizontal component of the gravitational force.
- Set F_c = F_h and solve for v.
Step 6: Use the relationship between velocity and frequency to determine the frequency of revolution.
- The velocity of an object moving in a circle is related to its frequency by the formula v = 2πr * f, where r is the radius of the circle and f is the frequency of revolution.
- Rearrange the equation to solve for f.
By following these steps, we can find the frequency of revolution of the mass.