Rotational Kinematics - angle between force and velocity

[trinkleb]

Here is the problem I am working on. I have found answers for all of them except part (f), which is the one I need help with. I will report the answers I have so far:

A classic 1957 Chevrolet Corvette of mass 1240 kg starts from rest and speeds up with a constant tangential acceleration of 2.00 m/s^2 on a circular test track of radius 60.0 m. Treat the car as a particle.

(a) What is its angular acceleration? --> 0.0333 rad/s²
(b) What is its angular speed 6.00 s after it starts? --> 0.2 rad/s
(c) What is its radial acceleration at this time? --> 2.4 rad/s²
(d) (I'll skip this one, since it's just a sketching problem)
(e) What are the magnitudes of the total acceleration and net force for the car at this time? --> a_tot = 3.12 m/s² and ΣF = 3874 N

(f) What angle do the total acceleration and net force make with the car's velocity at this time?

I'm wondering if I should use one of the rotational kinematics equations, but I'm still not sure how to go about it. Any ideas would be helpful. Thank you!

 

[K S Thurm]

Use your kinematics equations and substitute the variables. v=omega, alpha=a or acceleration, time=time, delta(x)=delta(theta).

Then solve as though kinematics. Use correct units: rad/s rad/s^2 instead of m/s etc. If you need more help, let me know.

 

[trinkleb]

This is what I did using a kinematics equation:

Θ = Θ_o + ω_ozt + 1/2(α_zt²

Since it starts from rest,

Θ = 1/2(0.0333 rad/s²)(6 s)²
Θ = 0.594 rad = 107°

This is not the right answer. Does Θ really represent the angle between the acceleration and the velocity? I thought it was just the angular displacement, in which I can't see a correlation between that and the acceleration and velocity angle.

 

[K S Thurm]

Delta Theta is Angular displacement. Starts from rest wi=0. Time=6.00s. Alpha=0.0333 rad/s^2. You now have 3 variables. Solve for wf.

Use wf=wi+alpha(time) -----------> wf=0 + (0.0333 rad/s^2)(6.00sec) That should result in your angular speed of 0.1998 or 0.2rad/s

Foe reg kinematics Vf=Vi+at

 

[K S Thurm]

Oh sorry, for part (f) try

Delta Theta = ((wi+wf)/2 ))(time) wi+wf, divide by 2, multiply by time

 

[trinkleb]

Alright, so when I use that equation I get:

ΔΘ = ½(ω_o + ω_f)(t)
ΔΘ = ½(0 + 0.2 rad/s)(6 s) = 0.6 rad = 34.4°

The book says it should be 50.2°, which doesn't make sense to me. Is there a certain concept that I might be missing which could be keeping me from getting the right answer?

 

[K S Thurm]

You need to use Arctan(ar/at). Try looking up Linear Acceleration in rigid body rotation in your textbook. atan=dv/dt = (r)(dw/dt)=ra

Tan^-1(2.40/2.00) =50.19 deg

 

[trinkleb]

Oh wow, that makes a lot of sense. Thank you so much!