# A wheel has a coonstant rotational acceleration of 3.0 rad/s^2. During a certain 4.0 s interval, it turns through an angle of 120 rad. Assuming that the wheel starts from rest, how long is it in motion at the start of the 4s interval.

I just need help in getting started. I know that the 3.0 rad/s^2 is the alpha roational acceleration and 120 rad is the theta. I do not have though rotational velocity.

Thank you

The way you ask the question does not make sense. At the START of the 4s interval, it is at rest, and you are asking "how long is it in motion?" At the start of the interval, it has not been in motion at all. You have already told us how long it is in motion, so why are you asking "how long"?

Perhaps you meant to ask how large an angle it turns through AT THE END OF 4 seconds. In that case,

Total angle of rotation = (1/2)*alpha* t^2

The displacement from t=0 is not 120, but 120 plus the displacement starting at the four second period

totaldisplacement=initialdisplacement + initalangularspeed*time + 1/2 angular acceleration*time^2

Starting from rest...

120+ initialdisplacement= +0+ 1/2*3 (t )^2

solve for t. this is the time from starting to the end of the four second period.

Now, initial displacement is 1/2*3(t-4)^2

so

120+1/2*3(t-4)^2=1/2*3 (t )^2

solve for t. You are looking for t-4 in this problem.

check my thinking.

## Haha, sorry for the confusion with the question! Let's break it down to make it easier to understand.

We know that the rotational acceleration (alpha) of the wheel is 3.0 rad/s^2 and the angle of rotation (theta) is 120 rad. We are trying to find out how long the wheel has been in motion at the start of the 4s interval.

First, let's calculate the time it takes for the wheel to reach the angle of 120 rad using the formula:

theta = 0.5 * alpha * t^2

120 = 0.5 * 3.0 * t^2

Now, solve for t using some algebra:

240 = 3.0 * t^2

t^2 = 80

t = √80

t ≈ 8.94 seconds

So, at the end of the 4s interval, the wheel has been in motion for approximately 8.94 seconds.

Now, to find out how long it has been in motion at the start of the 4s interval, we need to subtract 4 seconds from the total time:

Time at the start of the 4s interval = 8.94 - 4

Time at the start of the 4s interval ≈ 4.94 seconds

Therefore, the wheel has been in motion for approximately 4.94 seconds at the start of the 4s interval.

## Your initial understanding is correct: the given rotational acceleration is 3.0 rad/s^2 and the angle of rotation is 120 rad. However, you are right in pointing out that the question is a bit confusing in asking for "how long" the wheel is in motion at the start of the 4s interval, as it starts from rest.

To find the time it takes for the wheel to rotate through an angle of 120 rad, we can use the kinematic equations for rotational motion:

θ = ωi*t + (1/2) * α * t^2

where θ is the angle of rotation, ωi is the initial angular velocity (which is zero in this case), α is the angular acceleration, and t is the time.

In this case, we know that θ = 120 rad and α = 3.0 rad/s^2. Since the wheel starts from rest (ωi = 0), we can simplify the equation to:

120 rad = (1/2) * 3.0 rad/s^2 * t^2

Now, we can solve this equation to find the time it takes for the wheel to rotate through 120 rad:

240 = 3 * t^2

t^2 = 80

t = √80 ≈ 8.94 s

So it takes approximately 8.94 seconds for the wheel to rotate through an angle of 120 rad.

However, your clarification about the "start of the 4s interval" suggests that you may be looking for the time it takes for the wheel to start rotating. In that case, the wheel starts from rest and reaches the 120 rad rotation at 4 seconds. Therefore, it is in motion for the entire 4-second interval.

Please let me know if you need further clarification or if I can assist you with anything else.

## To start solving the problem, we can use the kinematic equation for rotational motion:

θ = ω₀t + (1/2)αt²

Where:

- θ is the angle of rotation (120 rad in this case),

- ω₀ is the initial angular velocity (0 rad/s because the wheel starts from rest),

- α is the rotational acceleration (3.0 rad/s²),

- t is the time interval (4.0 s in this case).

To find how long the wheel is in motion at the start of the 4.0 s interval, we need to determine the value of t when θ = 0 (since it starts from rest).

Since we know the values of θ, ω₀, α, and t, we can substitute them into the equation:

0 = 0 + (1/2)(3.0)(t)²

Simplifying the equation gives us:

0 = 1.5t²

This equation tells us that the wheel is at rest at the start of the 4.0 s interval. Therefore, it is not in motion.

If you meant to ask how large an angle the wheel turns through AT THE END OF the 4.0-second interval, we need to solve the equation with the given values:

120 rad = 0 + (1/2)(3.0)(4.0)²

Simplifying the equation gives us:

120 = 6t²

Dividing both sides by 6 gives:

20 = t²

Taking the square root of both sides gives:

t ≈ ±4.47 s

The negative root doesn't make physical sense in this context, so we only consider the positive root. Therefore, the wheel is in motion for approximately 4.47 seconds at the end of the 4.0 s interval.

Please let me know if anything is unclear or if you have any further questions!