Question

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?

Answer 1

5/12

Step-by-step 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