An oscillator consists of a block attached to a spring (k = 299 N/m). At some time t, the position (measured from the system's equilibrium location), velocity, and acceleration of the block are x = 0.0817 m, v = -16.9 m/s, and a = -108 m/s2. Calculate (a) the frequency (in Hz) of oscillation, (b) the mass of the block, and (c) the amplitude of the motion
To find the answers to these questions, we can use the equations of motion for simple harmonic oscillators.
(a) Frequency (in Hz) of Oscillation:
The frequency of oscillation can be found using the formula:
f = 1 / T
where T is the period of oscillation. We can calculate the period using the equation:
T = 2π √(m / k)
Here, m is the mass of the block, and k is the spring constant.
(b) Mass of the Block:
The mass of the block can be found using the equation:
m = (a / k)
(c) Amplitude of Motion:
The amplitude of motion can be determined from the given position of the block, which is the maximum displacement from the equilibrium position.
Let's solve the problem step-by-step:
(a) Frequency of Oscillation:
To find the frequency, we need to calculate the period (T) first. The formula for the period is:
T = 2π √(m / k)
Given:
k = 299 N/m
From the given information, we don't have the mass (m) or the period (T) directly, but we can calculate them using the available information.
First, let's calculate the period (T):
T = 2π √(m / k)
Since we don't have the mass (m) directly, we can calculate it using the equation:
m = (a / k)
m = (-108 m/s^2) / (299 N/m)
m = -0.3612 kg
Now, let's calculate the period (T):
T = 2π √(-0.3612 kg / 299 N/m)
T ≈ 2.145 s
Finally, let's find the frequency (f):
f = 1 / T
f = 1 / 2.145
f ≈ 0.465 Hz
So, the frequency of oscillation is approximately 0.465 Hz.
(b) Mass of the Block:
Using the equation:
m = (a / k)
m = (-108 m/s^2) / (299 N/m)
m = -0.3612 kg
So, the mass of the block is approximately 0.3612 kg.
(c) Amplitude of Motion:
The amplitude of motion is given as the displacement from equilibrium position, which is x = 0.0817 m.
Hence, the amplitude of motion is 0.0817 m.