A body of unknown mass is attached to an ideal spring with k=120 n/m. it is found to vibrate with a frequency of 4 Hz. find the period the angular frequency and the mass of the body
To find the period, angular frequency, and mass of the body, we can use the equations related to the motion of a mass-spring system.
1. To find the period:
The period (T) is the time taken for one complete oscillation or vibration. It is the reciprocal of the frequency (f). The formula for the period is:
T = 1 / f
Since the frequency (f) in this case is given as 4 Hz, we can substitute it into the equation to find the period:
T = 1 / 4 = 0.25 seconds
Therefore, the period of the vibration is 0.25 seconds.
2. To find the angular frequency:
The angular frequency (ω) is a measure of how quickly an object moves through its oscillations. It is related to the frequency by the formula:
ω = 2πf
Where π is approximately equal to 3.14159.
Substituting the given frequency of 4 Hz into the equation, we can find the angular frequency:
ω = 2 * 3.14159 * 4 = 25.13274 radians/second
Therefore, the angular frequency of the vibration is approximately 25.13274 radians/second.
3. To find the mass of the body:
To find the mass of the body (m), we can use the relationship between the angular frequency (ω) and the spring constant (k):
ω = √(k / m)
Squaring both sides of the equation gives:
ω² = k / m
Rearranging the equation to solve for mass (m), we get:
m = k / ω²
Substituting the given spring constant (k = 120 N/m) and angular frequency (ω ≈ 25.13274 radians/second) into the equation, we can calculate the mass:
m = 120 / (25.13274)² ≈ 0.19 kg
Therefore, the mass of the body is approximately 0.19 kg.