Question

Problem 1.3 The pendulum in Figure 1.1(a) swings with a displacement amplitude a. If its starting point from rest is (a) r=a (b)x= -a find the different values of the phase constant for the solutions x = a sin(wr+ ) x = a cos(wr+ ) x = a sin ( wo) r=a cosw- For each of the different values of , find the values of wit at which the pendulum swings through the positions * = +a/V2 x=a/2

please also include full explanation and write in paper. thankyou

Answer 1

(a) --

$x= a sin (\omega t \pm \phi )$   

• at
  a C= /2
  
• $x = \frac{a}{\sqrt{2}} = a sin (\omega t \pm \phi )$
• 
  sin(wt ) N
  
• $(\omega t \pm \phi ) = \frac{\pi }{4}$
• $\phi = \frac{\pi }{4}\mp \omega t$ ......................(answer)

at $x =\frac{a}{2}$

• $x =\frac{a}{2} = a sin (\omega t \pm \phi)$
• $\omega t \pm \phi= \frac{\pi}{6}$
• $\phi= \frac{\pi}{6} \mp \omega t$ ..................(answer)

​​​​​​​NOW FOR $x= a cos (\omega t \pm \phi )$

• at $x=\frac{a}{\sqrt{2}}$
• $x= a cos (\omega t \pm \phi ) =\frac{a}{\sqrt{2}}$
• $\phi =\frac{\pi }{4} \mp \omega t$ ....................(Answer)
• at $x =\frac{a}{2 }$
• $x= a cos (\omega t \pm \phi ) = \frac{a}{2}$
• $\phi = \frac{\pi}{3} \mp \omega t$ .................(answer)

()b

• $\omega ^{2} = \frac{g}{l}$
• $T = 2\pi \sqrt{\frac{l}{g}}$​​​​​​​