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 calculate the frequency of oscillation (part a), we can start by using the formula for the frequency of a simple harmonic oscillator:
f = 1 / (2π√(k / m))
Where:
f is the frequency,
k is the spring constant,
m is the mass of the block.
To calculate the mass of the block (part b), we can rearrange the above formula:
m = k / (4π²f²)
To calculate the amplitude of the motion (part c), we need to consider that at any point in the oscillation, the maximum displacement from the equilibrium position is the amplitude.
Let's calculate each part step by step:
(a) Frequency (f):
Given:
k = 299 N/m
Using the given equation:
f = 1 / (2π√(k / m))
To solve this equation, we need to find the value of the mass (m) first.
(b) Mass (m):
Given:
k = 299 N/m
f (which we'll calculate)
Using the rearranged formula:
m = k / (4π²f²)
To solve this equation, we need to find the value of the frequency (f) first.
(c) Amplitude:
The amplitude (A) is given by the maximum displacement from the equilibrium position, which can be found from the position (x) value:
Given:
x = 0.0817 m
The amplitude is simply the absolute value of x:
A = |x|
Now, let's calculate each part one by one.