Question

Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t = 0) position of the particle, the radius of the circle and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case. (x is in cm and t it is in s).a. x = -2 sin (3t + π/3)b. x = cos (π/6 - t)c. x = 3 sin (2πt + π/4)d. x = 2 cos πt

Answer 1

we know, standard equation of simple harmonic motion is given by, $x=Acos\left(\frac{2\pi}{T}t\pm\Theta\right)$
where A is amplitude, T is time period , $\Theta$ is phase angle ,

a. x = -2sin(3t + π/3)
x = 2cos(π/2 + 3t + π/3) [ as we know, cos(π/2 + R) = -sinR]
x = 2cos(3t + 5π/6)
hence, amplitude = 2, angular velocity , $\omega=\frac{2\pi}{T}=3rad/s$ , phase angle , $\Theta=\frac{5\pi}{6}$
The motion of the particle can be plotted as shown in fig. 10(a).

b. x = cos(π/6 - t) = cos(t - π/6) [ as cos(-R) = cosR]
compare with standard equation.
amplitude, A = 1 , angular velocity, $\omega=1rad/s$, phase angle , $\Theta$ = -30°
The motion of the particle can be plotted as shown in fig. 10(b).

c. x = 3sin(2πt + π/4)
x = -3cos(π/2 + 2πt + π/4)
x = -3cos(2πt + 3π/4)
compare with standard equation,
amplitude, A = 3cm [ here negative sign doesn't include ]
angular velocity, $\omega$ = 2π rad/s
and phase angle , $\Theta$ = 3π/4
The motion of the particle can be plotted as shown in fig. 10(c).

d. x = 2cosπt
x = 2cos(πt + 0)
amplitude, A = 2 rad/s
angular velocity, $\omega$ = π rad/s
phase angle, $\Theta$ = 0°
The motion of the particle can be plotted as shown in fig. 10(d)

Answer 2

Answer:

Given, Mass of the spring balance, m = 50 kg,

Length of the scale, y = 20 cm = 0.2 m,

Period of oscillation,T = 0.60 s

We know, F = ky [ from spring force ]

And F = mg [ from Newton's 2nd law, weight = mass × acceleration due to gravity.]

so, mg = ky

k = mg/y

= 50 × 9.8/0.2 N/m

= 2450 N/m

Now, time period in case of spring is given by,

so, T² = 4π²m/k => m = T²k/4π²

Put, T = 0.6 s , k = 2450 Nm

so, m = (0.6)² × 2450/4(3.14)²

m = 22.36 kg