How Do You Calculate Angular Acceleration and Speed from a Position Function?

[Fihzix]

Homework Statement

The angular position of a point on the rim of a 18.7 cm rotating wheel is given by θ(t) = 4.7 t2 − 7.2 t +9.7, where θ is measured in radians and t is measured in seconds.

What is the instantaneous angular acceleration α of the point at time t = 6 s?
What is the instantaneous tangential (not radial!) acceleration a of the point at time t =6 s?
What is the instantaneous angular velocity ω of the point at time t = 9 s?
What is the instantaneous speed v of the point at time t = 9 s?
What is the average angular speed ωav of the point over the time interval starting time t = 6 s and ending at the time t = 9 s?
Through what angular displacement Δθ does the wheel turn during this time?

The Attempt at a Solution

I get the angular position of 135.7 radian at 6 seconds but I do not understand where to go from there. I cannot just use those for angular velocity/acceleration right?

Thanks.

 

[rl.bhat]

If θ is given, how to find the angular velocity and angular acceleration.
Can you find them by differentiating θ(t)
What is relation between linear velocity and angular velocity?

 

[Fihzix]

Sorry, I was just introduced to this and I am super confused.

 

[rl.bhat]

Angular position of the wheel is given as θ(t) = 4.7 t2 − 7.2 t +9.7
The angular velocity = ω = d[θ(t)] /dt
The angular acceleration = α = d(ω)/dt
linear velocity v = ω*R
linear acceleration = a = Rα

 

[Fihzix]

I am able to get the last two questions involving average angular speed and displacement because I can simply plug in the numbers but I don't know how to find the instantaneous values. I know how to find the speed up to that point but not specifically at that instant.

 

[ace99]

hi, i too have a similar question to that, i was able to get the instantaneous angular velocity but doing derivatives then just plugin in the point, but i don't understand how to get the instantaneous angular acceleration and instantaneous tangential acceleration.

 

[rl.bhat]

hi, i too have a similar question to that, i was able to get the instantaneous angular velocity but doing derivatives then just plugin in the point, but i don't understand how to get the instantaneous angular acceleration and instantaneous tangential acceleration.

$$Angular acceleration \alpha = \frac{d^{2}\theta}{dt^{2}}$$

Tangential acceleration = Rα.

 

[ace99]

$$Angular acceleration \alpha = \frac{d^{2}\theta}{dt^{2}}$$

Tangential acceleration = Rα.

sorry to be a bother, but i still don't quite understand that formula. is it another derivative?

 

[rl.bhat]

Instantaneous angular velocity ω = dθ/dt
Instantaneous angular acceleration = α = dω/dt

 

[Fihzix]

I am also still confused about that formula. (surprise, surprise)

 

[rl.bhat]

In he problem θ(t) = 4.7*t^2 - 7.2*t + 9.7
Can you find the derivative of θ with respect to t?

 

[Fihzix]

I am still awaiting the derivative lecture in calculus. I expect this is why I am unable to continue in physics.

 

[rl.bhat]

Sorry. Without the knowledge of derivative you cannot solve this problem.

 

[Fihzix]

Thanks for your help, this makes much more sense to me now. I guess I can't do my physics assignment though. haha

 

[rl.bhat]

If you are very much particular about the physics assignment, why can't you open your maths book and go through basic rules of derivative?
Just one rule is sufficient.
If y = x^2, then dy/dx = 2x.
And if y = x, then dy/dx = 1.

 

[ace99]

If you are very much particular about the physics assignment, why can't you open your maths book and go through basic rules of derivative?
Just one rule is sufficient.
If y = x^2, then dy/dx = 2x.
And if y = x, then dy/dx = 1.

so what's the rule for d^2y/dx^2?

as in $$Angular acceleration \alpha = \frac{d^{2}\theta}{dt^{2}}$$

 

[rl.bhat]

so what's the rule for d^2y/dx^2?

as in $$Angular acceleration \alpha = \frac{d^{2}\theta}{dt^{2}}$$

If θ = t^2
then dθ/dt = 2t
and d^2(θ)/dt^2 = 2