Rotation problem - centrifugal forces

[henryc09]

Homework Statement

An object sliding along a smooth horizontal ruler that I am holding will fly off as a result of the centrifugal force if I am rotating. Write down Newton's 2nd law for the sliding object as viewed by me in the rotating frame whilst moving with angular velocity w, and verify that it is satisfied if the distance r from the rotation axis at time t is:

r(t)=Acosh(wt) + Bsinh(wt)

where A and B are arbitrary constants.

Homework Equations

The Attempt at a Solution

Well the only force causing acceleration in your frame of reference is the centrifugal force which is mw²r which is equal to md²r/dt². Cancel the mass but then I don't know where to go from here. I guess you solve the differential equation but I'm not sure how to do this. Any ideas?

 

[anathema]

Thank you for your interesting question. I would like to clarify a few points before answering your question. First, the concept of "centrifugal force" is actually a fictitious force that arises in a rotating frame of reference. In reality, there is only one force acting on the object, which is the centripetal force directed towards the center of rotation. Second, Newton's second law states that the net force on an object is equal to its mass times its acceleration. This law applies to both inertial and non-inertial frames of reference.

Now, let's consider the situation you described. In a rotating frame of reference, an object sliding along a smooth horizontal ruler will experience a centripetal force directed towards the center of rotation. This force is given by Fc = mrw^2, where m is the mass of the object, r is the distance from the center of rotation, and w is the angular velocity. This force is always directed towards the center of rotation, regardless of the position of the object on the ruler.

Using Newton's second law, we can write the equation of motion for the object as:

Fnet = mrw^2 = ma

where a is the acceleration of the object. Since we are in a rotating frame of reference, we need to consider the acceleration in this frame, which is given by:

a = d^2r/dt^2

Substituting the given expression for r(t), we get:

d^2r/dt^2 = -Aw^2sinh(wt) + Bw^2cosh(wt)

We can see that this equation satisfies Newton's second law, as the net force on the object is equal to its mass times its acceleration.

To solve this differential equation, we can use the method of undetermined coefficients. Assuming the solution takes the form r(t) = Acosh(wt) + Bsinh(wt), we can substitute it into the equation and solve for A and B. This will give us the position of the object as a function of time.

In conclusion, Newton's second law is satisfied in a rotating frame of reference, and we can use it to determine the motion of an object sliding along a ruler held by a rotating observer. I hope this helps answer your question. If you have any further doubts, please feel free to ask. Keep on exploring and learning![Your

 

[kerimek]

You are correct in identifying the centrifugal force as the only force causing acceleration in this frame of reference. Using Newton's second law, F=ma, and setting the force equal to the mass times the second derivative of the position, we get:

mw^2r = md^2r/dt^2

Since we are given the equation for r(t), we can take the second derivative with respect to time to verify that it satisfies the above equation:

d^2r/dt^2 = Aw^2cosh(wt) + Bw^2sinh(wt)

Substituting this into the equation for Newton's second law, we get:

mw^2(Acosh(wt) + Bsinh(wt)) = m(Aw^2cosh(wt) + Bw^2sinh(wt))

The mass cancels out, leaving us with:

w^2(Acosh(wt) + Bsinh(wt)) = Aw^2cosh(wt) + Bw^2sinh(wt)

This equation is satisfied, showing that the equation for r(t) does indeed satisfy Newton's second law in the rotating frame of reference. To solve the differential equation, we can use the initial conditions to determine the values of A and B.