An oscillator consists of a block of mass 512 g connected to a spring. When set into oscillation with amplitude 34.7 cm, it is observed to repeat its motion every 0.484 s. Find (a) the period, (b) the frequency, (c) the angular frequency, (d) the force constant, (e) the maximum speed, and ( f ) the maximum force exerted on the block.

To analyze the motion of the oscillator you've described, we can break down the problem into several parts. Let's tackle each component step by step, using the information provided: a mass of 512 g, an amplitude of 34.7 cm, and a period of 0.484 s.

Calculating the Period

The period (T) is the time it takes for one complete cycle of oscillation. You've already provided this value as 0.484 seconds. So:

• Period (T): 0.484 s

Determining the Frequency

Frequency (f) is the number of cycles per second and is the reciprocal of the period. We can calculate it using the formula:

f = 1/T

Substituting the period we found:

f = 1/0.484 s ≈ 2.07 Hz

• Frequency (f): 2.07 Hz

Finding the Angular Frequency

The angular frequency (ω) relates to the frequency and is given by the formula:

ω = 2πf

Plugging in the frequency we calculated:

ω = 2π(2.07 Hz) ≈ 13.00 rad/s

• Angular Frequency (ω): 13.00 rad/s

Calculating the Force Constant

The force constant (k) of the spring can be found using the formula:

k = mω²

First, we need to convert the mass from grams to kilograms:

m = 512 g = 0.512 kg

Now substituting the values:

k = 0.512 kg * (13.00 rad/s)² ≈ 87.06 N/m

• Force Constant (k): 87.06 N/m

Maximum Speed Calculation

The maximum speed (v_max) of the block can be calculated using the formula:

v_max = Aω

Where A is the amplitude in meters:

A = 34.7 cm = 0.347 m

Now substituting the values:

v_max = 0.347 m * 13.00 rad/s ≈ 4.52 m/s

• Maximum Speed (v_max): 4.52 m/s

Maximum Force Exerted on the Block

The maximum force (F_max) exerted on the block can be calculated using Hooke's Law:

F_max = kA

Substituting the values we have:

F_max = 87.06 N/m * 0.347 m ≈ 30.19 N

• Maximum Force (F_max): 30.19 N

Summary of Results

To summarize, here are the key results from our calculations:

• Period (T): 0.484 s
• Frequency (f): 2.07 Hz
• Angular Frequency (ω): 13.00 rad/s
• Force Constant (k): 87.06 N/m
• Maximum Speed (v_max): 4.52 m/s
• Maximum Force (F_max): 30.19 N

This breakdown should give you a clear understanding of the oscillator's behavior and the relationships between its various properties. If you have any further questions or need clarification on any of these concepts, feel free to ask!