The angular displacement in radians is given by: θ=8√(t^2+2)
Find the angular velocity and angular acceleration at t = 0.25 s
To find the angular velocity and angular acceleration at t = 0.25s, we first need to find the derivative of the angular displacement function with respect to time.
Given: θ = 8√(t^2 + 2)
Step 1: Find the derivative of θ with respect to t. This will give us the angular velocity function.
dθ/dt = d/dt (8√(t^2 + 2))
= 8 * d/dt (√(t^2 + 2))
= 8 * 1/2 * (t^2 + 2)^(-1/2) * 2t
= 8t / √(t^2 + 2)
Therefore, the angular velocity function is: ω(t) = 8t / √(t^2 + 2)
Step 2: Find the derivative of ω with respect to t. This will give us the angular acceleration function.
dω/dt = d/dt (8t / √(t^2 + 2))
= 8 * (d/dt t / √(t^2 + 2) + t * d/dt (1 / √(t^2 + 2)))
= 8 * (1 / √(t^2 + 2) - t * (1 / 2) * (t^2 + 2)^(-3/2) * 2t)
= 8 * (1 / √(t^2 + 2) - t^2 / (t^2 + 2)^(3/2))
Therefore, the angular acceleration function is: α(t) = 8 * (1 / √(t^2 + 2) - t^2 / (t^2 + 2)^(3/2))
Now, to find the angular velocity and angular acceleration at t = 0.25s, we substitute t = 0.25 into the respective functions:
For angular velocity: ω(0.25) = 8(0.25) / √((0.25)^2 + 2)
For angular acceleration: α(0.25) = 8 * (1 / √((0.25)^2 + 2) - (0.25)^2 / ((0.25)^2 + 2)^(3/2))
Calculating these values will give you the angular velocity and angular acceleration at t = 0.25s.