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Chapter 9: Problem 151

If \(y, u\), and \(a\) represent displacement, velocity, and acceleration at any instant for a particle executing SHM, which of the following statements are true? (A) \(v\) and \(y\) may have same direction. (B) \(v\) and \(a\) have same direction twice in each cycle. (C) \(a\) and \(y\) may have same direction. (D) \(a\) and \(v\) never have same direction.

Short Answer

Expert verified
The true statements are (A) \(v\) and \(y\) may have the same direction and (B) \(v\) and \(a\) have the same direction twice in each cycle.

Step by step solution

01

Recall the formulas for SHM

For a particle executing SHM, we know the following formulas: Displacement: \( y(t) = A \sin(\omega t + \phi) \) Velocity: \( u(t) = A \omega \cos(\omega t + \phi) \) Acceleration: \( a(t) = -A \omega^2 \sin(\omega t + \phi) \) Where \(A\) represents the amplitude, \(\omega\) represents the angular frequency, \(t\) represents time, and \(\phi\) represents the phase angle.
02

Analyze Statement A

(A) \(v\) and \(y\) may have the same direction. From the formulas for displacement and velocity, we can see that the signs of the displacement and velocity can be the same or opposite depending on the value of \(\omega t + \phi\). If \(\omega t + \phi\) lies in the first or third quadrants, \(y\) and \(u\) will have the same direction. Therefore, statement A is true.
03

Analyze Statement B

(B) \(v\) and \(a\) have the same direction twice in each cycle. From the formulas for velocity and acceleration, we can see that \(u\) and \(a\) have the same sign when \(\omega t + \phi\) lies in the second and fourth quadrants. Since there are two such instances in a full cycle (2蟺), statement B is true.
04

Analyze Statement C

(C) \(a\) and \(y\) may have the same direction. Since acceleration is negative of the product of angular frequency squared and displacement, if we look at the formulas for displacement and acceleration, \(a\) and \(y\) have opposite directions, whenever displacement is positive acceleration is negative, and vice versa. Therefore, statement C is false.
05

Analyze Statement D

(D) \(a\) and \(v\) never have the same direction. Referring back to Step 3, we already know that the velocity and acceleration can have the same direction twice in each cycle, and therefore statement D is false. So, the true statements are (A) and (B).

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