# Finding the angular velocity of a rotating arm

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In summary, the conversation discusses the application of three forces (normal, elastic, and centrifugal) on a slider moving along the x-axis. The elastic force is found to be equal to 2R, and the equation N-Fe+m(ω^2R)=0 is solved to find the value of ω, which is √2.5. The same equation is used for problem B, but with N set to 0. For problem C, a new term R_eq is introduced to represent the radius at equilibrium, and the equation is solved for ω. It is noted that the value of ω affects the centrifugal force, which is m(ω^2r). Overall, the concepts discussed are found
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Homework Statement
There's an arm which is ##2R## long and rotates with constant angular velocity on a horizontal plane. There's also a slider block with mass ##0.4 kg##. This slider is pushed against a support ##A## by a spring of constant ##2 N/m## and natural length ##2R##.
A) Suppose that the force exerted by the support is half the force exerted by the elastic force, find the angular velocity of the system.
B) Find the minimum angular velocity so that the slider doesn't interact with the support.
C) If the angular velocity found previously is doubled, which would be the equilibrium position with respect to the arm?
D) Which would be the difference if the angular velocity is now rotating the other way?
Relevant Equations
Newton's equations
A) The slider experiments three forces, all of them are on the ##x## axis (considering ##x## axis as the axis aligned with the arm): Normal force (exerted by the support), elastic force and centrifugal force, which is ##m.(\omega^2 r)##

Elastic force is equal to
##Fe=-k \delta =-2 (R-2R)=2R## and therefore ##N=R##. Then, Newton equations would be
##N-Fe+m (\omega^2 R)=0##. So, solving for ##\omega## I get:
##m.\omega^2 R=R##, and then, replacing the value of the mass we get ##\omega = \sqrt{2.5}##

B) For B we solve the same equation but set ##N=0##.

C) For C, we solve
##k(R_{eq} -2R)+m\omega^2 R_{eq} =0##, where ##R_{eq}## is the radius for which the system is in equilibrium.

D) The angular velocity affect the centrifugal force, which is ##m.(\omega^2 r)##. As ##\omega## is squared, the sign doesn't change anything.
Have I solved the problem correctly? I don't know if my concepts are correct

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Like Tony Stark said:
C) For C, we solve
##k(R_{eq} -2R)+m\omega^2 R_{eq} =0##, where ##R_{eq}## is the radius for which the system is in equilibrium.
Where omega is what value?

Detail: a statement like ##R = N## hurts my eyes: the dimensions have disappeared. Write e.g:$$N = -{1\over2} k(R-2R) \quad \Rightarrow \\ N + k(R - 2R) = {1\over2} k(R-2R) = {1\over2}kR\quad \Rightarrow\\ N + k(R - 2R) +m (\omega^2 R)=0 \ \ \Leftrightarrow\ \ \omega^2 = {1\over2} {kR\over mR} \quad \Rightarrow \\ \omega = \sqrt {2.5}\ \text {rad/s }$$

haruspex
haruspex said:
Where omega is what value?
The double of the calculated in B. Sorry I didn't make any difference, but I didn't want to write ##\omega_2## or something like that. I thought it was easy to understand just like that

BvU said:

Detail: a statement like ##R = N## hurts my eyes: the dimensions have disappeared. Write e.g:$$N = -{1\over2} k(R-2R) \quad \Rightarrow \\ N + k(R - 2R) = {1\over2} k(R-2R) = {1\over2}kR\quad \Rightarrow\\ N + k(R - 2R) +m (\omega^2 R)=0 \ \ \Leftrightarrow\ \ \omega^2 = {1\over2} {kR\over mR} \quad \Rightarrow \\ \omega = \sqrt {2.5}\ \text {rad/s }$$
Thanks and sorry! It looks ugly to me too, but it was messy to me to write all the units in the post (I write them when doing the homework, obviously)

BvU
Practice makes perfect

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## What is angular velocity?

Angular velocity is a measure of how fast an object is rotating around a fixed axis. It is usually expressed in radians per second (rad/s) or degrees per second (deg/s).

## How do you calculate angular velocity?

Angular velocity can be calculated by dividing the change in angle (in radians or degrees) by the change in time. The formula for angular velocity is: ω = Δθ/Δt, where ω is angular velocity, Δθ is change in angle, and Δt is change in time.

## What is the difference between angular velocity and linear velocity?

Angular velocity measures the rate of change of angular displacement, while linear velocity measures the rate of change of linear displacement. In other words, angular velocity is a rotational speed, while linear velocity is a straight line speed.

## How does the length of the rotating arm affect the angular velocity?

The length of the rotating arm does not affect the angular velocity. Angular velocity depends on the rate of change of angle, not the length of the arm. However, the length of the arm can affect the linear velocity of a point on the arm.

## Can angular velocity be negative?

Yes, angular velocity can be negative. A negative angular velocity indicates that the object is rotating in the opposite direction of the positive direction. For example, a clockwise rotation would have a positive angular velocity, while a counterclockwise rotation would have a negative angular velocity.

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