Question

For a simple harmonic oscillator, answer yes or no to the following questions. (a) Can the quantities position and velocity have the same sign? (b) Can velocity and acceleration have the same sign? (c) Can position and acceleration have the same sign?

Short answer

(a)Yes

(b) Yes

(c)No

Step-by-step solution

Step 1: Position and initial velocity of the particle

$T=2\pi \sqrt{\frac{m}{k}}$

$T=$Tension in the string

$k=$Spring constant

$m=$Mass of object

Step 2(a): Find the quantities position and velocity have the same sign

Yes. In simple harmonic motion, one-half of the time, the velocity is in the same direction as the displacement away from equilibrium.

Step 3(b): Find the velocity and acceleration have the same sign

Yes. Velocity and acceleration are in the same direction half the time.

Step 4(c): Find the position and acceleration have the same sign

No. The spring force and, therefore, the acceleration are always opposite to the position vector, and never in the same direction.