At t=0 a grinding wheel has an angular velocity of 24.0rad/s . It has a constant angular acceleration of 35.0rad/s2 until a circuit breaker trips at time t = 1.90s . From then on, it turns through an angle 436rad as it coasts to a stop at constant angular acceleration.

Through what total angle did the wheel turn between t=0 and the time it stopped?

At what time did it stop?

What was its acceleration as it slowed down?

v = Vi + a t

Vend = 24 + 35 (1.9)

angle at 1.9 s = 1.9(Vi+Vend)/2

total angle = 436 + angle at 1.9 s

time at Vend = 1.9 + 436/Vend

To find the total angle turned by the wheel, we need to calculate the angles covered during different time intervals.

Step 1: Calculate the angle covered during the acceleration phase (0 to 1.9 seconds):
Using the equation: θ = ω₀t + 0.5αt²
where θ is the angle covered, ω₀ is the initial angular velocity, t is the time, and α is the angular acceleration.

θ₁ = ω₀t + 0.5αt²
θ₁ = (24 rad/s)(1.9 s) + 0.5(35 rad/s²)(1.9 s)²
θ₁ = 45.6 rad + 0.5(35 rad/s²)(3.61 s²)
θ₁ = 45.6 rad + 0.5(35 rad/s²)(3.61 s²)
θ₁ = 45.6 rad + 62.0075 rad
θ₁ ≈ 107.61 rad

Step 2: Calculate the remaining angle covered during the coasting phase (time after 1.9 seconds):
The coasting phase has a constant angular acceleration, but the time at which it stops is unknown. Let's represent the time the wheel stops as t.

Using the equation: θ = ω₀t + 0.5αt²
where θ is the angle covered, ω₀ is the initial angular velocity, t is the time, and α is the angular acceleration.

θ₂ = ω₀t + 0.5αt²
θ₂ = 0 rad + 0.5αt²
θ₂ = 0.5αt²

Step 3: Substitute values and solve for the remaining angle:
θ₁ + θ₂ = 107.61 rad + 0.5αt²
θ₂ = 436 rad - 107.61 rad
0.5αt² = 328.39 rad

Step 4: Calculate the time at which the wheel stops:
Using the equation: θ = ω₀t + 0.5αt²
θ₂ = 0.5αt²
328.39 rad = 0.5(α)t²

We know that the initial angular velocity is 0 rad/s (since it stopped), and α is the angular acceleration during the coasting phase. We can substitute those values:

328.39 rad = 0.5(α)t²
656.78 rad = αt²

Step 5: Calculate the acceleration (α) during coasting phase:
We can rearrange the equation to isolate α:

α = (656.78 rad)/(t²)

Since the wheel coasts to a stop, the final angular velocity is 0 rad/s. This means the acceleration will be negative.

Now we need to solve for t.

To calculate t, we need to find the positive root of the equation by rearranging and using the quadratic formula:

t² = (656.78 rad)/(α)
t = sqrt((656.78 rad)/(α))

Given that α = -αcoasting, we can substitute -α for α in the equation:

t = sqrt((656.78 rad)/(-αcoasting))

This gives us the time at which the wheel stops.

Therefore, the total angle turned by the wheel between t=0 and the time it stopped is approximately 436 rad.

The wheel stops at a time of approximately t = sqrt((656.78 rad)/(-αcoasting)).

The acceleration (α) as it slows down is calculated using the formula α = (656.78 rad)/(t²), where t is the time at which the wheel stops.

To find the total angle the wheel turned between t=0 and the time it stopped, we need to calculate the sum of the angles during different time intervals.

1. Compute the angle covered during the first interval of constant angular acceleration until the circuit breaker trips.

The formula to calculate the angle during this period is given by:
θ1 = ω1 * t + (1/2) * α * t^2
where:
θ1 is the angle covered during this interval
ω1 is the initial angular velocity of the wheel at t = 0
α is the angular acceleration
t is the time at which the circuit breaker trips

Plugging in the values:
ω1 = 24.0 rad/s
α = 35.0 rad/s^2
t = 1.90 s

θ1 = (24.0 rad/s) * (1.90 s) + (1/2) * (35.0 rad/s^2) * (1.90 s)^2

Calculate θ1.

2. Compute the angle covered during the second interval as the wheel coasts to a stop at constant angular acceleration.

The formula to calculate the angle during this period is given by:
θ2 = ω2^2 / (2 * α)
where:
θ2 is the angle covered during this interval
ω2 is the final angular velocity of the wheel at the time it stops
α is the angular acceleration

Since the wheel comes to a stop at the end of this interval, ω2 would be 0.

θ2 = 0^2 / (2 * α)

Calculate θ2.

3. Sum up θ1 and θ2 to find the total angle covered.

Total angle = θ1 + θ2

Calculate the total angle covered by summing up the values obtained in steps 1 and 2.

To find the time when the wheel stops, we can use the following relation:

ω2 = ω1 + α * t_stop
where:
ω2 is the final angular velocity of the wheel at the time it stops
ω1 is the initial angular velocity of the wheel at t = 0
α is the angular acceleration
t_stop is the time at which the wheel stops

We can rearrange the equation to solve for t_stop:

t_stop = (ω2 - ω1) / α

Plugging in the values:
ω2 = 0 (since the wheel stops)
ω1 = 24.0 rad/s
α = 35.0 rad/s^2

Calculate t_stop.

To find the acceleration as the wheel slows down, we can use the formula:

a = α

Since the angular acceleration α is constant throughout the motion, the acceleration as the wheel slows down is equal to α.

Calculate the acceleration.