Circular motion lab problem

[corollary]

Homework Statement

For a conical pendulum:

Construct a free-body diagram for the pendulum and derive an expression for the acceleration in terms of the gravitational field strength g, the length L of the pendulum, the radius r of the circular path of the pendulum, and the frequency f of oscillation. (Hint - first derive an expression for tanθ the angle between the string and the vertical.)

Homework Equations

a=v^2/r
a=Δv/Δt
a=ƩF/m

The Attempt at a Solution

Have done free body diagram, can't figure out how to make equation.

 

[haruspex]

There is a specific format you are supposed to use on this forum.
What do you have for the forces - just describe them, don't worry about a diagram.
What do you have for the acceleration?

 

[corollary]

Thank you for replying; didn't see I had to use that form.
As for forces there is the force of gravity on the pendulum mass (downwards), and the tension in the string.
As for the acceleration - I have no numerical value for it. The question needs no actual values - just variables to make the equation (a in terms of g, L, r, and f).

 

[haruspex]

Thank you for replying; didn't see I had to use that form.
As for forces there is the force of gravity on the pendulum mass (downwards), and the tension in the string.
As for the acceleration - I have no numerical value for it. The question needs no actual values - just variables to make the equation (a in terms of g, L, r, and f).

True, but you do know the direction the acceleration has to be in. So in what directions would you resolve the forces, and what equations result?

 

[Nickie]

As a scientist, it is important to be able to accurately and clearly communicate your findings and solutions. In this case, it seems that you have successfully constructed a free-body diagram for the conical pendulum, but are struggling with deriving an expression for the acceleration. Here is a step-by-step guide to help you:

1. Start by drawing a diagram of the conical pendulum, labeling all the forces acting on it. These include the tension force (T) from the string, the weight force (mg), and the centripetal force (Fc) acting towards the center of the circular path.

2. Use trigonometry to find the relationship between the angle θ and the length of the pendulum (L) and the radius of the circular path (r). This will give you an expression for tanθ.

3. Next, consider the forces acting on the pendulum in the horizontal and vertical directions. In the horizontal direction, the only force is the tension force (T), which is equal to the centripetal force (Fc). This can be written as T=Fc.

4. In the vertical direction, there are two forces - the weight force (mg) and the vertical component of the tension force (Tsinθ). These two forces must be balanced for the pendulum to remain in circular motion. This can be written as mg=Tsinθ.

5. Substitute the expression for T from step 3 into the equation from step 4. This will give you an equation with only one unknown variable - θ.

6. Use the equation a=v^2/r to relate the acceleration (a) to the velocity (v) and the radius (r) of the circular path.

7. Finally, use the equation a=Δv/Δt to relate the acceleration (a) to the frequency (f) of oscillation and the velocity (v).

8. Combine the equations from steps 6 and 7 to eliminate v and solve for the acceleration (a) in terms of g, L, r, and f.

I hope this helps you to derive the expression for the acceleration in the conical pendulum. Remember to always take your time and carefully work through each step, and don't hesitate to ask for assistance if you need it. Good luck!