Understanding F=ma: How to Prove It with Conical Pendulum Equations"

In summary, The task is to verify F=ma using equations from a conical pendulum lab. The instructor provided instructions to divide the equations and show the relationship between mass, length of the string, radius, and period. However, the mention of including the length of the string and using trigonometry has caused confusion. The key is to remember the circular motion equations and use trigonometry to incorporate the length of the string.
  • #1
wanu
1
0
It's not a problem, it's a proof. The trouble being that I'm not entirely sure what I'm supposed to be proving, which is why I'm getting so confused. Our instructor told us to verify F=ma using the equations that we got from a conical pendulum lab. When further prompted, he said to divide them (which I took to mean finding the tangent) There was also mention of showing the relationship between m (mass), l (length of the string the mass was on), R (the radius of the circle), and P (the period. (In retrospect, that seems to be the most important instruction.)



Although I keep confusing myself, this is what I think I know:
where T refers to the Tension force
T_x =Tsin(theta T) =mv^2/R
T_y= Tcos(theta T)=mg
and from those I got tan(theta)=v^2/Rg=4pi^2R/Pg



And that's where I get stuck, because I'm not entirely sure how to include l in the relationship, and if I do it should have to do with h (the height of the theoretical triangle), right? So if that's true, how do I take that into account and what should I be doing with the tan(theta)?

Any help or guidance would be greatly appreciated-- I really just want to be able to wrap my head around this.
 
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  • #2
If you have a look at the circular motion equations you can probably substitute in for acceleration to get all the quantities you want. Just remember your trigonometry as that is where you will get the length of the string into the equation.
 
  • #3


I can understand your confusion and frustration with this problem. It can be challenging to prove a concept using equations and experimental data, especially when there are multiple variables involved. However, with some careful analysis and understanding of the principles involved, we can work through this together.

Firstly, we need to understand the physical concept behind F=ma. This equation, also known as Newton's second law of motion, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In other words, the larger the force applied to an object, the greater its acceleration will be, and the more massive the object is, the smaller its acceleration will be.

Now, let's look at the equations you have derived from the conical pendulum lab. As you correctly stated, the tension force (T) can be broken down into its horizontal and vertical components, T_x and T_y, respectively. The horizontal component, T_x, is responsible for providing the centripetal force that keeps the mass moving in a circular path. This force is equal to mv^2/R, where m is the mass of the object, v is its velocity, and R is the radius of the circle.

On the other hand, the vertical component, T_y, balances the weight of the object (mg) to keep it in equilibrium. From these two equations, we can see that the tension force (T) is directly related to the mass (m) and the radius of the circle (R).

Now, let's bring in the concept of the period (P), which is the time it takes for the mass to complete one full revolution. From your equation, tan(theta)=v^2/Rg=4pi^2R/Pg, we can see that the period is also related to the radius and the acceleration due to gravity (g).

But how does length (l) come into play? Well, as you correctly mentioned, it is related to the height (h) of the theoretical triangle formed by the string, the mass, and the vertical component of the tension force. This height, in turn, affects the angle (theta) at which the string makes with the vertical, which is what you are trying to determine.

To fully understand the relationship between all these variables, we need to use some trigonometry. From the diagram of the conical pendulum, we can see that the length of the
 

1. What is a conical pendulum?

A conical pendulum is a type of pendulum that consists of a mass attached to a string or rod, which is suspended from a fixed point and swings in a circular motion.

2. How does a conical pendulum work?

A conical pendulum works by utilizing the force of gravity and the tension in the string or rod to keep the mass moving in a circular path. The circular motion is caused by the horizontal component of the tension force, while the vertical component balances out the force of gravity.

3. What factors affect the motion of a conical pendulum?

The motion of a conical pendulum is affected by the length of the string or rod, the mass of the object, and the angle at which the pendulum is released. Other factors such as air resistance and friction can also play a role.

4. What is the difference between a conical pendulum and a simple pendulum?

A conical pendulum differs from a simple pendulum in that its motion is not limited to a back-and-forth swing, but rather a circular motion. The forces acting on a conical pendulum are also more complex, as it involves both the tension force and the force of gravity.

5. How is a conical pendulum used in real life?

Conical pendulums are often used in physics demonstrations and experiments to illustrate concepts such as circular motion and centripetal force. They can also be found in some clocks and timekeeping devices, as well as in amusement park rides.

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