Explaining Angular Frequency: \theta, m, R, r & k

[suspenc3]

https://www.physicsforums.com/showthread.php?p=846415

Can anyone explain this a little bit more?

I found these 2 formulas.
$$\tau=-k \theta$$

How do I relate $$\theta$$ with m, R, r & k? (In my problem I am not working with numbers, i need to find an expression for angular frequency in terms of R, r, m, k,

$$\tau=I \alpha$$ where $$I=MR^2$$

therefore $$\alpha = \frac{-k_t\theta}{MR^2}$$

Do I just resolve $$\theta$$ into its horizontal component? That wouldn't really get rid of $$\theta$$ though...Confused
Thanks

 

[OlderDan]

https://www.physicsforums.com/showthread.php?p=846415

Can anyone explain this a little bit more?

I found these 2 formulas.
$$\tau=-k \theta$$

How do I relate $$\theta$$ with m, R, r & k? (In my problem I am not working with numbers, i need to find an expression for angular frequency in terms of R, r, m, k,

$$\tau=I \alpha$$ where $$I=MR^2$$

therefore $$\alpha = \frac{-k_t\theta}{MR^2}$$

Do I just resolve $$\theta$$ into its horizontal component? That wouldn't really get rid of $$\theta$$ though...Confused
Thanks

From the diagram in the link you posted you should be able to write an expression for the elongation (or compression) of the spring Δx in terms of the small angular displacement of the wheel from the equilibrium position. The force applied at the point of connection between the spring and the wheel is proportional to Δx. The torque about the axle of the wheel is the result of that force applied at distance r from the axis of rotation.

The "k" in your equation above is not necessarily the spring constant of the spring in the diagram. You need to find the proportionality constant between torque and angular displacement.

 

[m2003]

for the question. Angular frequency, denoted by \omega, is a measure of how fast an object is rotating or oscillating. It is defined as the rate of change of angular displacement with respect to time. In other words, it is the amount of rotation an object experiences per unit time.

In the formulas you have provided, \theta represents the angular displacement, which is the change in the angle of rotation. This can be related to the other variables in the following way:

- m represents the mass of the object
- R represents the distance of the object from the center of rotation
- r represents the radius of the object itself
- k represents the spring constant in a spring-mass system

In the formula \tau=-k\theta, \tau represents the torque applied to the object. This can be thought of as a twisting force that causes the object to rotate. The negative sign indicates that the direction of rotation is opposite to the direction of the torque.

In the second formula, \tau=I\alpha, I represents the moment of inertia, which is a measure of an object's resistance to rotational motion. It is equal to the product of the mass and the square of the distance from the center of rotation (MR^2). \alpha represents the angular acceleration, which is the rate of change of angular velocity with respect to time.

To relate \theta to these variables, we can use the formula \alpha=\frac{\tau}{I}. Substituting the values for \tau and I from the previous formulas, we get \alpha=\frac{-k\theta}{MR^2}. This can also be written as \omega=\sqrt{\frac{k}{MR^2}}, which gives the expression for angular frequency in terms of the other variables.

To answer your last question, resolving \theta into its horizontal component may not be necessary as long as you use the correct values for the other variables. I hope this explanation helps to clear up any confusion.