Question

Gives the x-t plot of a particle executing one-dimensional simple harmonic motion. Give the signs of position, velocity and acceleration variables of the particle at t = 0.3 s, 1.2 s, -1.2 s.

Answer 1

Let the maximum amplitude of the sine wave be A, where A is positive. We can see that the particle obeys the equation

x = – A sin(2πt/T)

Where T = 2 s = period of the sine wave

∴ position = x = – A sin(πt)

velocity – v = dx/dt = – A_πcos(πt)

acceleration = a = dv/dt = A_π²sin(πt)

At t = 0.3 s

x = – A sin (0.3_π) = negative

v = – A cos (0.3_π) = negative

a = A_π² sin(0.3_π) = positive

Since sin(0.3_π) > 0 and cos(0.3_π) > 0

At t = 1.2 s

x = – A sin(1.2_π) = – A sin(1.2_π) = positive

v = – A_π cos(1.2_π) = – A cos(1.2_π) = positive

a = A_π² sin(1.2_π) = negative

Since sin(1.2_π) < 0 and cos(1.2π) < 0

At t = – 1.2 s

x = – A sin(- 1.2_π) = A sin(1.2_π)= negative

v = – A it cos(- 1.2_π) = – A_πcos(1.2π) = positive.

a = A_π² sin(- 1.2_π) = – A_π2sin(1.2_π) = positive

Since sin(-θ) = – sinθt cos(-θ) = cosθ sin(1.2_π) < 0 and cos(1.2_π) < 0