To find the angle through which the wheel rotates in the interval t = 0s to t = 37s, we need to break down the problem into different time intervals and calculate the net angle rotated in each interval.
Let's start by calculating the angular velocity at t = 17s, when the acceleration abruptly changes to 0 rad/s^2. We know that the angular acceleration is constant until then, so we can use the formula:
ω = ω₀ + αt
Where:
ω = angular velocity at time t
ω₀ = initial angular velocity
α = angular acceleration
t = time
Given:
ω₀ = 0 rad/s (starting from rest)
α = given constant angular acceleration
t = 17s
We can calculate:
ω = 0 + α * 17
Now we have the angular velocity at t = 17s. Next, we need to calculate the angle rotated from t = 0s to t = 17s.
We can use the formula:
θ = ω₀t + (1/2)αt²
Where:
θ = angle rotated in time t
ω₀ = initial angular velocity
α = angular acceleration
t = time
Given:
ω₀ = 0 rad/s (starting from rest)
α = given constant angular acceleration
t = 17s
We can calculate:
θ₁ = 0 * 17 + (1/2) * α * (17)²
θ₁ is the angle rotated from t = 0s to t = 17s.
Now, let's calculate the angular velocity at t = 37s, where the total time is given:
Given:
t = 37s
ω = 5.5 rad/s (at t = 1.6s)
We can rearrange the formula ω = ω₀ + αt to solve for angular acceleration:
α = (ω - ω₀) / t
We know ω₀ = 0 rad/s (starting from rest), so we have:
α = (5.5 - 0) / 1.6
Now, we have the angular acceleration for the interval t = 1.6s to t = 37s. To calculate the angle rotated in this interval, we can use the formula:
θ = ω₀t + (1/2)αt²
Where:
θ = angle rotated in time t
ω₀ = initial angular velocity
α = angular acceleration
t = time
Given:
ω₀ = 5.5 rad/s (at t = 1.6s)
α = calculated angular acceleration
t = 37s - 1.6s = 35.4s
We can calculate:
θ₂ = 5.5 * 35.4 + (1/2) * α * (35.4)²
θ₂ is the angle rotated from t = 1.6s to t = 37s.
To find the total angle rotated from t = 0s to t = 37s, we can sum up the angles calculated in both intervals:
θ_total = θ₁ + θ₂
Now, you can use the calculated values to find the angle through which the wheel rotates in the given time interval t = 0s to t = 37s.