Answer:
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
