The motion of a particle is described by x = 10 sin (πt + π/3), where x is in meters and t is in seconds. At what time in seconds is the potential energy equal to the kinetic energy?

5/12

Explanation:

Given

x = 10 sin (πt + π/3)

v = distance/time

So, v = dx/dt

Differentiating x with respect to t

v = 10π cos (πt + π/3)

Also,

½kx² = ½mv²

Substituting values for x and v in the above equation

½k (10sin (πt + π/3)) ² = ½m (10πcos (πt + π/3)) ²

Divide through by ½

k (10sin (πt + π/3)) ² = m (10πcos (πt + π/3)) ²

Open both bracket

100ksin² (πt + π/3) = 100mπ²cos² (πt + π/3)

Divide through by 100

ksin² (πt + π/3) = mπ²cos² (πt + π/3)

Divide through by kcos² (πt + π/3)

ksin² (πt + π/3) : kcos² (πt + π/3) = mπ²cos² (πt + π/3) : kcos² (πt + π/3)

tan² (πt + π/3) = mπ²/k

tan² (πt + π/3) = (m/k) π²

But w² = k/m and w = 2π/T

(2π/T) ² = k/m

(2π) ²/T² = k/m

1/T² = k/m : (2π) ²

1/T² = k/m * (2π) ²

T² = m (2π) ²/k

From the Question, T is when πt = 2π or T = 2

Substitute 2 for T in the above equation

2² = m (2π) ²/k

4 = m (2π) ²/k

4 = 4π²m/k

m/k = 1/π²

(m/k) π² = 1

Remember that tan² (πt + π/3) = (m/k) π²

So, tan² (πt + π/3) = 1

This gives

πt + π/3 = 45° = π/4

πt + π/3 = π/4

Divide through by π

t + ⅓ = ¼

t = ¼ - ⅓

t = - 1/12 - - - Negative

Using the second quadrant

πt + π/3 = 3π/4

Divide through by π

t + ⅓ = ¾

t = ¾ - ⅓

t = 5/12