Question

two particle A and B executing SHM along same straight line with same amplitude and same mean position A start its motion from mean position and move toward positive extreme while B starts from negative extreme position Angular frequency of A is Wa and that of B is Wb choose the incorrect statment
A)if Wa =2Wb then when they meet first their velocity will be zero
B)ifWa>2Wb then when they meet first time their velocity are in same direction
C)ifWa<2Wb then when they meet their velocity will be in same direction
D)their velocity when they meet does not depend on W

Answer 1

Answer on Question #73380, Physics / Mechanics | Relativity

Question. Two particle $A$ and $B$ executing $SHM$ along same straight line with same amplitude and same mean position. $A$ starts its motion from mean position and moves toward positive extreme while $B$ starts from negative extreme position. Angular frequency of $A$ is $\omega_A$ and that of $B$ is $\omega_B$ choose the incorrect statement

A) if $\omega_A = 2\omega_B$ then when they meet first their velocity will be zero.

B) if $\omega_A > 2\omega_B$ then when they meet first time their velocity are in same direction.

C) if $\omega_A < 2\omega_B$ then when they meet their velocity will be in same direction.

D) their velocity when they meet does not depend on $\omega$.

Solution.

A) if $\omega_A = 2\omega_B$ then when they meet first their velocity will be zero.

Assume that

$$x_A(t) = \sin(\omega_A t),$$ $$x_B(t) = \sin\left(\omega_B t - \frac{\pi}{2}\right)$$

So

$$v_A(t) = \frac{dx_A(t)}{dt} = \omega_A \cos(\omega_A t),$$ $$v_B(t) = \frac{dx_B(t)}{dt} = \omega_B \cos\left(\omega_B t - \frac{\pi}{2}\right),$$

If $x_A(t) = x_B(t)$ and $\omega_A = 2\omega_B$ then

$$\sin(\omega_A t) = \sin\left(\omega_B t - \frac{\pi}{2}\right) \rightarrow \sin(2\omega_B t) = \sin\left(\omega_B t - \frac{\pi}{2}\right) \rightarrow$$ $$\sin(2\omega_B t) - \sin\left(\omega_B t - \frac{\pi}{2}\right) = 0$$

The solution of this equation is $\omega_B t = \frac{2\pi n}{3} + \frac{\pi}{2}$, $n \in \mathbb{Z}$. Hence

$$v_A(t) = \omega_A \cos(\omega_A t) = 2\omega_B \cos(2\omega_B t) = -2\omega_B.$$ $$v_B(t) = \omega_B \cos\left(\frac{\pi}{2} - \frac{\pi}{2}\right) = \omega_B.$$

In fig. $\omega_{B} = 3$ rad/s² and $\omega_{A} = 6$ rad/s².

Answer. The statement is incorrect.

B) If $\omega_{A} > 2\omega_{B}$ then when they meet first time their velocity are in same direction.

Answer. The statement is incorrect.

C) if $\omega_{A} < 2\omega_{B}$ then when they meet their velocity will be in same direction.

Answer. The statement is correct.

D) Their velocity when they meet does not depend on $\omega$ .

The velocity when they meet depend on $\omega$ (see A)).

Answer. The statement is incorrect.

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