Problem in a non-inertial frame

[coolguy]

hey guys!
i have a problem in physics which i could'nt solve:

a car is moving with constant linear acceleration a along horizontal x-axis. A solid sphere of mass M and radius R is found rolling without slipping on the horizontal floor of the car in the same direction as seen from an inertial frame outside the car.What is the acceleration of the sphere in the inertial frame?

Could anyone help me with it please?

thanks.

 

[enricfemi]

How the sphere rolling?
I mean its acceleration in car'frame.
Generally speaking,its acceleration in the inertial frame =car'acceleration+sphere's acceleration in car'frame
It's Gallilus theory

 

[coolguy]

the sphere is rolling without slipping...thats all that has been given in the problem...does a pseudo force come into play??

 

[Doc Al]

No need for pseudoforces. Hint: Since it rolls without slipping, what must be the acceleration of the bottom surface of the sphere? Apply Newton's 2nd law for both translational and rotational motion.

 

[coolguy]

i tried. but i am not getting it.could u be more precise with it please.
thanks.

since it is rolling without slipping,static friction comes into play.
let a(obs) be the acceleration observed in the inertial frame.let f be the force of friction.then
f*r = 2/5 m*(r^2)*a(obs)*r

now applying Newton's 2nd law
m*(a - a(obs)) = f

this step is a bit confusing and am getting the wrong answer.

 

[OlderDan]

i tried. but i am not getting it.could u be more precise with it please.
thanks.

since it is rolling without slipping,static friction comes into play.
let a(obs) be the acceleration observed in the inertial frame.let f be the force of friction.then
f*r = 2/5 m*(r^2)*a(obs)*r

now applying Newton's 2nd law
m*(a - a(obs)) = f

this step is a bit confusing and am getting the wrong answer.

What Doc Al is suggesting is that you work entirely in the inertial observers frame of reference. The force of friction will accelerate the CM of the ball, and it will produce a torque that gives the ball angular acceleration. You need to find the relationship between the linear and angular accelerations based on the no-slipping condition.

An analogous problem you may have seen is a disk with a string wound around it laying flat on a frictionless table. What happens if the string is pulled horizontally with a constant force?

 

[Doc Al]

since it is rolling without slipping,static friction comes into play.

Good.

let a(obs) be the acceleration observed in the inertial frame.let f be the force of friction.then

Careful with this. The observed acceleration of the car is given as "a". Are you using "a(obs)" to be the acceleration of the sphere? (The acceleration of its center of mass, I presume?)

f*r = 2/5 m*(r^2)*a(obs)*r

Another place to be careful. What you need here is the sphere's angular acceleration about its center of mass.

now applying Newton's 2nd law
m*(a - a(obs)) = f

I guess I don't know what "a(obs)" means.

The key, as OlderDan says, is this:

You need to find the relationship between the linear and angular accelerations based on the no-slipping condition.

Hint: Try to express the angular acceleration of the sphere in terms of the accelerations of the sphere and the truck.

 

[coolguy]

the a(obs) here is the required acceleration of the sphere with respect to the inertial frame.
are the two relations i have mentioned then correct?
please help with the second one.with that clear the problem will be simple to solve.

 

[Doc Al]

the a(obs) here is the required acceleration of the sphere with respect to the inertial frame.
are the two relations i have mentioned then correct?

If that's the case, then neither is correct.

please help with the second one.with that clear the problem will be simple to solve.

OK.

now applying Newton's 2nd law
m*(a - a(obs)) = f

This is Newton's 2nd law for translation. You have called the force "f" and the acceleration "a(obs)". Rewrite it.

(But you need to redo your first equation also.)

 

[chaoseverlasting]

Can we use f_r=\mu mg and substitute that in the equation
f_r=\frac{I\alpha}{r} and then use

a_o-r\alpha=a where a_o is the acceleration of the CM of the sphere?

 

[coolguy]

thanks buddies.the answer correctly comes out to be a(obs) = (2/7)*a.

the only problem was due to the relative acceleration between the sphere and the car.
also friction need not be equal to (mu)*m*g.the force of friction gets canceled out.

thanks anyway.

 

[Doc Al]

the answer correctly comes out to be a(obs) = (2/7)*a.

Good!

the only problem was due to the relative acceleration between the sphere and the car.

Right; it's needed to get the proper angular acceleration of the sphere about its center.

also friction need not be equal to (mu)*m*g.the force of friction gets canceled out.

Exactly right.

 

[enricfemi]

the answer correctly comes out to be a(obs) = (2/7)*a.

I just computed it.But come out to be (2/5)*R^2*a.And I suppose the correct answer should include R.Is there something wrong with the correct answer?
my course:
friction force=Ia=(2/5)*m*R*R*a
inertial force=friction force(It maybe relative with a course called theory mechanics )

 

[Doc Al]

I just computed it.But come out to be (2/5)*R^2*a.And I suppose the correct answer should include R.Is there something wrong with the correct answer?

Your answer is not even dimensionally correct. The correct answer is independent of R.

my course:
friction force=Ia=(2/5)*m*R*R*a

Try using: Torque = I alpha

 

[enricfemi]

Indeed!
But apply this the answer is 2/5*a?

 

[enricfemi]

"Try using: Torque = I alpha"
Indeed!
By apply this the answer is 2/5*a?

 

[Doc Al]

But apply this the answer is 2/5*a?

What are you using for alpha? Express it in terms of "a" and "a(obs)". Hint: It's not a/r. Alpha is the angular acceleration of the sphere about its center of mass.

 

[enricfemi]

My course:
friction force=I*alpla/R=I*a/(R^2)=(2/5)*m*a
And I supposed friction force =inertial force.

 

[Doc Al]

friction force=I*alpla/R=I*a/(R^2)=(2/5)*m*a

As I said earlier, alpha does not equal a/r. To find alpha, you need the acceleration of the sphere's surface with respect to its center.

And I supposed friction force =inertial force.

For some reason, you seem to want to view this from a non-inertial frame, which is OK. But if the inertial force equaled friction (the only two forces on the sphere), the sphere would be in translational equilibrium.