Cloverleaf highway interchange - determine car acceleration

[Alexanddros81]

Homework Statement

13.34 The curved portion of the cloverleaf highway interchange is defined by
$R^2=b^2sin2θ$, 0<=θ<=90deg. If a car travels along the curve at the constant speed v₀,
determine its acceleration at A

Homework Equations

The Attempt at a Solution

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Is this wrong?

 

[kuruman]

Check your expression for $\ddot{R}$. There should be a $\ddot{\theta}$ in it.​

 

[Marc Rindermann]

and also you're missing a $\dot \theta^2$.

 

[Alexanddros81]

Check your expression for ¨RR¨\ddot{R}. There should be a ¨θθ¨\ddot{\theta} in it.

Is the following correct:

and also you're missing a ˙θ2θ˙2\dot \theta^2

I cannot see where I am missing the $\dot θ^2$

 

[Marc Rindermann]

first of all you can check whether your equations are correct if you do a dimensional analysis.
Look at $\ddot R = 2b\dot\theta$

$[\ddot R] = ms^{-2}$
$[2b\dot\theta] = ms^{-1}$

So you see something is not quite right.

When you take the 2nd derivative of $\dot R = b\cos(2\theta)(\sin(2\theta))^{-1/2}\dot\theta$ you need to remember that the derivative of both $\cos(2\theta)$ and $(\sin(2\theta))^{-1/2}$ produces another $\dot\theta$.

 

[Alexanddros81]

Can you check this?