Non-uniform circular motion and tangential acceleration

[Symstar]

Homework Statement

An object of mass m is constrained to move in a circle of radius r. Its tangential acceleration as a function of time is given by $$a_{tan} = b + ct^2$$, where b and c are constants.

A) If $$v = v_0$$ at t = 0, determine the tangential component of the force, $$F_{\tan }$$, acting on the object at any time t > 0.
Express your answer in terms of the variables m, r, $$v_0$$, b, and c.

B) Determine the radial component of the force $$F_{\rm{R}}$$.
Express your answer in terms of the variables m, r, $$v_0$$, b, t, and c.

Homework Equations

$$a_{tan} = b + ct^2$$
$$a_r=\tfrac{v^2}{r}$$
Newton's Laws

The Attempt at a Solution

A. was not a problem for me:
$$F_{\tan}=ma_{\tan}=m(b+ct^2)$$

For B.:
$$F_R=ma_r$$
$$a_r=\tfrac{v^2}{r}$$
It seems to make sense that because v is tangential speed we could use...
$$v(t)=v_0+a_{\tan}t=v_0+(b+ct^2)t$$
So that...
$$a_r=\frac{(v_0+(b+ct^2)t)^2}{r}$$
Finally giving...
$$F_R=m(\frac{(v_0+(b+ct^2)t)^2}{r}$$

Which is not correct. What did I do wrong?

 

[tiny-tim]

Welcome to PF!

Hi Symstar! Welcome to PF!

It seems to make sense that because v is tangential speed we could use...
$$v(t)=v_0+a_{\tan}t$$

That only works if a_tan is constant, doesn't it?

Hint: dv/dt = … ?

 

[Symstar]

Hi Symstar! Welcome to PF! That only works if a_tan is constant, doesn't it?

Hint: dv/dt = … ?

dv/dt = a_tan correct?

So would I need to integrate?
$$\int a_{tan} = \int b + ct^2$$
$$\frac{dv}{dt}= bt+\tfrac{1}{3}ct^3$$

And it seems logical in our case that +C would actually be +v₀

Which would end up giving me:
$$F_R=m\frac{(v_0+bt+\tfrac{1}{3}ct^3)^2}{r}$$

Which I just confirmed to be the correct answer... thanks for you your help tim.