# Another Bar Pendulum Question

• maverick280857
In summary, the conversation discusses the determination of the values of g and k for a bar pendulum using a straight line plot and experimental setup. The equations for the time period and moment of inertia are given, and it is mentioned that for a fixed time period, there are two values of d that can be measured. The equation for the moment of inertia around the center of mass is also mentioned. The conversation ends with a question about the validity of a certain approximation.

## Homework Statement

How can one determine from a straight line plot, the value of g (acceleration due to gravity) and k (radius of gyration) of a bar pendulum?

## Homework Equations

The time period of a bar pendulum is given by

$$T = 2\pi\sqrt{\frac{I}{Mgd}}$$

$$I = I_{0} + Md^2$$

where I is the moment of inertia about an axis passing through the pivot, d is the distance between the center of mass and the pivot, M is the mass of the rod.

In an experimental setup, we are varying d and measuring T for every chosen d.

## The Attempt at a Solution

Also,

$$I_{0} = Mk^2$$

So,

$$I = M(k^2 + d^2)$$

Hence,

$$T = 2\pi\sqrt{\frac{k^2 + d^2}{gd}}$$

For a fixed value of T, there are two values of d, and

$$d_{1} + d_{2} = \frac{4\pi^2(d_{1} + d_{2})}{T^2}$$

and

$$d_{1}d_{2} = k^2$$

But which straight line plot yields k and g? I can see that if k >> d, then we can say that T^2 is proportional to 1/d, but this would be a gross approximation, valid only for points very close to the center of mass.

Thanks
Cheers
vivek

Moment of inertia around a center of mass for perfect rod of length 2k is:

$$I_{0}=\frac{1}{12}m(2k)^2$$

How did you got this equation:

$$d_{1} + d_{2} = \frac{4\pi^2(d_{1} + d_{2})}{T^2}$$
?

Fixed walue of T? What two walues of $d$?

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## 1. What is a bar pendulum?

A bar pendulum is a physical system consisting of a rigid bar, suspended from a fixed point, with a mass attached to its end. It is used to demonstrate the principles of oscillation and simple harmonic motion.

## 2. How does a bar pendulum work?

A bar pendulum works by converting potential energy into kinetic energy as the mass attached to the end of the bar swings back and forth. The motion is controlled by the length of the bar and the force of gravity acting on the mass.

## 3. What factors affect the period of a bar pendulum?

The period of a bar pendulum is affected by the length of the bar, the mass of the attached object, and the acceleration due to gravity. The period is longer for longer bars, heavier masses, and higher gravity.

## 4. How can a bar pendulum be used in scientific experiments?

A bar pendulum can be used in scientific experiments to study the properties of simple harmonic motion, such as amplitude, frequency, and period. It can also be used to demonstrate the relationship between potential and kinetic energy.

## 5. What are the limitations of using a bar pendulum in experiments?

One limitation of using a bar pendulum in experiments is that it assumes a perfect system with no external forces or friction. In reality, there will always be some friction and other external factors that can affect the motion of the pendulum. Additionally, the assumptions of a simple harmonic motion may not hold true for all systems.