The angle through which a rotating wheel has turned in time t is given by \theta = a t - b t^2+ c t^4, where \theta is in radians and t in seconds.
What is the average angular velocity between t = 2.0 s and t =3.5 s?
Compute theta at t = 3.5 seconds and at t = 2.0 seconds. Divide the difference (the change in theta) by 1.5 seconds. The result will be the average angular velocity for that interval. It will depend upon the values of a, b and c.
To find the average angular velocity between t = 2.0 s and t = 3.5 s, we need to find the change in angle and the change in time between these two points.
Given:
\theta = a t - b t^2 + c t^4
Let's calculate the change in angle (\Delta\theta):
\Delta\theta = \theta_2 - \theta_1
Substituting the values of t = 3.5 s and t = 2.0 s:
\Delta\theta = (a(3.5) - b(3.5)^2 + c(3.5)^4) - (a(2.0) - b(2.0)^2 + c(2.0)^4)
Simplifying the equation:
\Delta\theta = (3.5a - 12.25b + 42.875c) - (2.0a - 4.0b + 16.0c)
\Delta\theta = 1.5a - 8.25b + 26.875c
Now, let's calculate the change in time (\Delta t):
\Delta t = t_2 - t_1
\Delta t = 3.5 - 2.0
\Delta t = 1.5
Finally, the average angular velocity (\omega_avg) is given by:
\omega_avg = \frac {\Delta\theta}{\Delta t}
\omega_avg = \frac {1.5a - 8.25b + 26.875c}{1.5}
Therefore, the average angular velocity between t = 2.0 s and t = 3.5 s is (1.5a - 8.25b + 26.875c) / 1.5.
To find the average angular velocity between t = 2.0 s and t = 3.5 s, we need to calculate the change in angle and divide it by the change in time.
First, we find the initial angle at t = 2.0 s by substituting t = 2.0 s into the expression for θ:
θ₁ = a(2.0) - b(2.0)^2 + c(2.0)^4
Next, we find the final angle at t = 3.5 s by substituting t = 3.5 s into the expression for θ:
θ₂ = a(3.5) - b(3.5)^2 + c(3.5)^4
Finally, we calculate the change in angle by subtracting the initial angle from the final angle:
Δθ = θ₂ - θ₁
To find the average angular velocity, we divide the change in angle by the change in time:
ω_avg = Δθ / Δt
Where Δt is the change in time, given by t₂ - t₁.
Now, you can substitute the given values of t₁, t₂, a, b, and c into the above equations to calculate the average angular velocity.