Question

1. two different springs, a and b, have weights hanging from them. both are pulled down and released, leading to harmonic motion as the weights move up and down. a student recorded the times when the weights were at the bottom and the top of the motion.

which spring has the higher frequency?
weight time at bottom time at top time at bottom time at top
a 0.0 s 0.5 s 1.0 s 1.5 s
b 0.0 s 1.0 s 2.0 s 3.0 s
spring a, 1.0 hz
spring a, 1.0 s
spring b, 2.0 hz
spring b, 2.0 s
clear all

Explanation:

Step1: Define period formula

The period \(T\) is the time for one - complete cycle. For a harmonic motion, if we consider the time from one bottom - position to the next bottom - position as one cycle.

Step2: Calculate period of Spring A

For Spring A, the time from \(t = 0.0\ s\) (first bottom) to \(t = 1.0\ s\) (second bottom) is \(T_A=1.0\ s\).

Step3: Calculate frequency of Spring A

The frequency \(f\) is related to the period by \(f=\frac{1}{T}\). So \(f_A=\frac{1}{T_A}=\frac{1}{1.0\ s}=1.0\ Hz\).

Step4: Calculate period of Spring B

For Spring B, the time from \(t = 0.0\ s\) (first bottom) to \(t = 2.0\ s\) (second bottom) is \(T_B = 2.0\ s\).

Step5: Calculate frequency of Spring B

Using \(f=\frac{1}{T}\), we get \(f_B=\frac{1}{T_B}=\frac{1}{2.0\ s}=0.5\ Hz\).

Step6: Compare frequencies

Since \(1.0\ Hz>0.5\ Hz\), Spring A has the higher frequency.

Answer:

Spring A, 1.0 Hz