14.10 In Exercise 14.9, let us take the position of mass when the spring is unstretched as x = 0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch (t = 0), the mass is

(a) At the mean position

(b) At the maximum stretched position

(c) At the maximum compressed position

In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

2 Views | Posted 5 months ago

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1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

5 months ago

The functions have the same frequency and amplitude, but different initial phases.

Distance travelled by the mass sideways, A = 2.0 cm

Force constant of the spring, k = 1200 N/m and mass, m = 3 kg

Angular frequency, $ω = \sqrt{\frac{k}{m}}$ = $\sqrt{\frac{1200}{3}}$ = 20 rad/s

(a) When mass is at the mean position, displacement x = Asin $ω t = 2 s i n 20 t$ cm

(b) At the maximum stretched position, the mass is towards extreme right. Hence the initial phase is $\frac{π}{2}$

x = A sin ( $ω t + \frac{π}{2})$ = 2sin (20t + $\frac{π}{2})$ = 2cos20t

(c) At the maximum compressed position, the mass is towards the extreme left

Hence, the initial phase is $\frac{3 π}{2}$

x = Asin ( $ω t + \frac{3 π}{2})$ = 2 cos20t

Hence, the f

...more

The functions have the same frequency and amplitude, but different initial phases.

Distance travelled by the mass sideways, A = 2.0 cm

Force constant of the spring, k = 1200 N/m and mass, m = 3 kg

Angular frequency, $ω = \sqrt{\frac{k}{m}}$ = $\sqrt{\frac{1200}{3}}$ = 20 rad/s

(a) When mass is at the mean position, displacement x = Asin $ω t = 2 s i n 20 t$ cm

(b) At the maximum stretched position, the mass is towards extreme right. Hence the initial phase is $\frac{π}{2}$

x = A sin ( $ω t + \frac{π}{2})$ = 2sin (20t + $\frac{π}{2})$ = 2cos20t

(c) At the maximum compressed position, the mass is towards the extreme left

Hence, the initial phase is $\frac{3 π}{2}$

x = Asin ( $ω t + \frac{3 π}{2})$ = 2 cos20t

Hence, the functions have the same frequency and amplitude, but different initial phases.

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The functions have the same frequency and amplitude, but different initial phases.Distance travelled by the mass sideways, A = 2.0 cmForce constant of the spring, k = 1200 N/m and mass, m = 3 kgAngular frequency,  <math><mi>ω</mi><mo>=</mo><mi></mi><msqrt><mrow><mfrac><mrow><mrow><mi>k</mi></mrow></mrow><mrow><mrow><mi>m</mi></mrow></mrow></mfrac></mrow></msqrt></math> = <math><msqrt><mrow><mfrac><mrow><mrow><mn>1200</mn></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></mfrac></mrow></msqrt></math> = 20 rad/s (a) When mass is at the mean position, displacement x = Asin <math><mi></mi><mi>ω</mi><mi>t</mi><mo>=</mo><mn>2</mn><mi>s</mi><mi>i</mi><mi>n</mi><mn>20</mn><mi>t</mi></math> cm  (b) At the maximum stretched position, the mass is towards extreme right. Hence the initial phase is <math><mfrac><mrow><mrow><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math>x = A sin ( <math><mi>ω</mi><mi>t</mi><mo>+</mo><mi></mi><mfrac><mrow><mrow><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mo>)</mo></math> = 2sin (20t + <math><mfrac><mrow><mrow><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mo>)</mo></math> = 2cos20t  (c) At the maximum compressed position, the mass is towards the extreme leftHence, the initial phase is <math><mfrac><mrow><mrow><mn>3</mn><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math>x = Asin ( <math><mi>ω</mi><mi>t</mi><mo>+</mo><mi></mi><mfrac><mrow><mrow><mn>3</mn><mi>π</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mo>)</mo></math> = 2 cos20tHence, the functions have the same frequency and amplitude, but different initial phases.

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