Toy car moving in a horizontal ring.

[peripatein]

Hi,

I sincerely hope one of you could please help me with the equations in the following problem.

Homework Statement

A toy car is forced to move along a circular ring placed on a table with no friction. The friction coefficient between the ring and the car is given as μ. At t=0 the car is moving along the inner side of the ring at a tangential velocity v_0.
I am asked for the velocity of the car wrt t.

Homework Equations

The Attempt at a Solution

I wrote down the following equations:
(1) N = mg
(2) μN - mv^2/r = ma_r
(3) v = v_0 + a_r*t

Are these equations correct? Is this the correct way to find v_t?

 

[peripatein]

Suggestion: supposing equations 1 and 2 above are correct, and 3 is irrelevant in this case, should v_t (velocity wrt time) rather be found by integration of a_r?

 

[tiny-tim]

hi peripatein!

A toy car is forced to move along a circular ring placed on a table with no friction. The friction coefficient between the ring and the car is given as μ.

i don't understand …

what is forcing the car? what is the magnitude of the force?

and are the car's wheels rolling? (because if so the friction will not slow the car, it will only help it to turn)

 

[peripatein]

I believe "forced" here is merely an indication that it keeps on moving within the ring. In any case, I do not think that detail should be given any attention in this case.
It is also not indicated whether the wheels are rolling.
Are the equations correct? How should v_t be found?

 

[tiny-tim]

if the car is rolling, and if there is no applied force, then the speed should be constant

 

[peripatein]

But then it would make no sense to ask for v_t. It is certainly for a reason stated that the car starts at t=0 with v_0.
Friction is the requisite force.

 

[tiny-tim]

if you don't know what's forcing the car to move in a circle (eg, is it a string, is it a rial, is it steerable front wheels?), then you don't know whether some of the friction is doing that, and so you don't know the component of friction in the tangential direction

(also don't forget that the question seems to say that the ring is free to rotate on the table)

 

[haruspex]

A toy car is forced to move along a circular ring placed on a table with no friction. The friction coefficient between the ring and the car is given as μ. At t=0 the car is moving along the inner side of the ring at a tangential velocity v_0.
I am asked for the velocity of the car wrt t.

I struggled to understand the set-up. The most sense I can make of it is that:
- the ring is fixed on the table
- the car is sliding/rolling on the table (doesn't matter which since there's no friction there) and its side is sliding against the ring.
So, first thing is to write down equations relating normal force from the ring, frictional retardation, tangential velocity and tangential acceleration.

 

[tiny-tim]

the ring is fixed on the table
- the car is sliding/rolling on the table (doesn't matter which since there's no friction there) and its side is sliding against the ring.

oh yes, that makes sense!

 

[peripatein]

Okay, which is basically what I was trying to do. Are the above equations correct? If not, could you please guide me how to reformulate them?

 

[peripatein]

In case I didn't make it clear, the ring is horizontally placed on the table. Moreover, please see attachment for, possibly, a better understanding of the set-up.

 

[haruspex]

(1) N = mg
(2) μN - mv^2/r = ma_r
(3) v = v_0 + a_r*t
Are these equations correct?

You need to define your terms before it is possible to judge. In particular, which force is N? N=mg makes it seem like it's the normal from the table, but then you have μN, which makes no sense because the table is frictionless.
I'm finding that with most of these mechanics questions the first advice needed is to list all the forces, assigning them unique identifiers, and determine the direction (and if appropriate the line of action) in which they act.

 

[tiny-tim]

hi peripatein!

(just got up :zzz:)

now everything looks clearer, let's examine this equation in some detail …

(2) μN - mv^2/r = ma_r

you obviously intend this to be the F = ma equation for the radial direction, but there are several things wrong with it …

i] there is no friction in the radial direction, is there?

ii] a_r is 0 (because r is constant)

iii] you haven't included the horizontal (radial) reaction force from the ring

you need to use F = ma and the centripetal acceleration to find the reaction force,

then use the reaction force to find the (tangential) friction force,

then use that to find the tangential deceleration​

start again

 

[peripatein]

Should it be:
(1) N=mv^2/r
(2) (Mu)N=ma_t
where a_t denotes the tangential deceleration

?

 

[tiny-tim]

yup!