A block attached to a spring oscillates in simple harmonic motion along the x-axis between the two extremes, x = 10 cm and x = 30 cm. The block goes from one extreme to the other in 0.25 s. Its amplitude and frequency are:
A = (30-10)/2 = 10 cm
the period is 2*.25 = .5 s
f = 1/.5s = 2 Hz
To determine the amplitude and frequency of the block's simple harmonic motion, we can use the given information about the two extremes and the time period.
1. Amplitude (A):
The amplitude of simple harmonic motion is the maximum displacement of the object from its equilibrium position. In this case, the block oscillates between x = 10 cm and x = 30 cm. The amplitude can be calculated as half the distance between the two extremes.
Amplitude (A) = (30 cm - 10 cm) / 2
Amplitude (A) = 20 cm / 2
Amplitude (A) = 10 cm
Therefore, the amplitude of the block's motion is 10 cm.
2. Time period (T):
The time period is the time taken for the block to complete one full oscillation or go from one extreme to the other and back. It is given as 0.25 s.
Time period (T) = 0.25 s
3. Frequency (f):
The frequency of simple harmonic motion is the number of oscillations per unit time. It is the reciprocal of the time period.
Frequency (f) = 1 / Time period (T)
Frequency (f) = 1 / 0.25 s
Frequency (f) = 4 Hz
Therefore, the frequency of the block's motion is 4 Hz.
To find the amplitude and frequency of the block's oscillation, we can use the formula for the simple harmonic motion:
x(t) = A * sin(ωt + φ)
where:
- x(t) is the displacement of the block at time t,
- A is the amplitude of the oscillation,
- ω is the angular frequency,
- φ is the phase constant.
Given that the block goes from x = 10 cm to x = 30 cm in 0.25 s, we can determine the amplitude as the half the difference between the two extremes:
Amplitude (A) = (30 cm - 10 cm) / 2 = 20 cm / 2 = 10 cm
The frequency (f) can be calculated using the formula:
f = 1 / T
where T is the period of the oscillation. The period is the time it takes for one complete cycle of the oscillation. In this case, we know the time it takes for the block to go from one extreme to the other (0.25 s). Hence, the period can be calculated as:
Period (T) = 2 * 0.25 s = 0.5 s
Therefore, the frequency is:
Frequency (f) = 1 / T = 1 / 0.5 s = 2 Hz
So, the amplitude of the block's motion is 10 cm and the frequency is 2 Hz.