Question

You want to find the moment of inertia of a complicated machine part about an axis through its center of mass. You suspend it from a wire along this axis. The wire has a torsion constant of 0.450 N \(\cdot\) m/rad. You twist the part a small amount about this axis and let it go, timing 165 oscillations in 265 s. What is its moment of inertia?

Short answer

The moment of inertia is approximately 0.0374 kg⋅m².

Step-by-step solution

Find the Period of Oscillation

We start by calculating the period (\( T \)) of a single oscillation of the system. The period is the total time divided by the number of oscillations. Here, it is given by:\[ T = \frac{265 \text{ s}}{165} \approx 1.6061 \text{ s} \]

Use the Formula for a Torsional Oscillator

The period of a torsional oscillator is related to the torsion constant (\( \kappa \)) and the moment of inertia (\( I \)) by the formula:\[ T = 2\pi \sqrt{\frac{I}{\kappa}} \]We know \( T = 1.6061 \text{ s} \) and \( \kappa = 0.450 \text{ N} \cdot \text{m/rad} \).

Solve for the Moment of Inertia

We rearrange the formula to solve for the moment of inertia (\( I \)):\[ I = \frac{T^2 \cdot \kappa}{4\pi^2} \]Plug in the values we know:\[ I = \frac{(1.6061)^2 \cdot 0.450}{4\pi^2} \approx 0.0374 \text{ kg} \cdot \text{m}^2 \]

Finalize the Answer

The moment of inertia of the machine part about the axis through its center of mass is approximately \( 0.0374 \text{ kg} \cdot \text{m}^2 \).

Key concepts

Torsional Oscillation

Torsional oscillation refers to the twisting motion of an object that is suspended by a wire or rod and rotates back and forth about its axis. This kind of motion is similar to the swinging of a pendulum, but instead of swinging, the object spins. The term "torsional" comes from the word "torsion," which means twisting. In the context of physics, torsional oscillation happens when an object twists and returns to its starting position repeatedly.

When the object is twisted slightly and released, it tends to rotate due to the torque provided by the twisting of the wire. This motion is periodic, meaning it happens at regular intervals. Torsional oscillations are common in mechanical systems, like in certain clocks or in the experiment of finding an object's moment of inertia. Understanding torsional oscillation is key to analyzing systems where rotational motion occurs due to the restoring torque.

Torsion Constant

The torsion constant (\( \kappa \)) is a crucial parameter when it comes to torsional oscillation systems. It represents how resistant the wire or rod is to being twisted. Essentially, it measures the stiffness or rigidity of the wire. It's similar to how a spring constant measures the stiffness of a coil spring.

A wire with a high torsion constant requires more torque to twist it by a given angle than a wire with a lower torsion constant. The unit for the torsion constant is typically Newton-meters per radian (\( \text{N} \cdot \text{m/rad} \)). In practical applications, if you know the torsion constant of a wire and the angle of twist, you can calculate the torque required to achieve that twist. This understanding helps in deriving and verifying the moment of inertia during an experiment involving oscillations as it is a part of the formula used to calculate the period of oscillation.

Calculation of Period of Oscillation

To calculate the period of oscillation (\( T \)), you need to know how long it takes for the system to complete one full cycle of motion. This period is essential to connecting the time-based measurement with the physical properties of the system like the torsion constant and the moment of inertia.

To find the period, divide the total time it takes for a set number of oscillations by the number of those oscillations. For example, if a system completes 165 oscillations in 265 seconds, the period (\( T \)) can be calculated as:

• \( T = \frac{265 \ \text{s}}{165} \approx 1.6061 \ \text{s} \)

Once the period is known, you can use it in formulas to uncover other properties such as moment of inertia. For torsional systems, the relationship between the period (\( T \)), torsion constant (\( \kappa \)), and moment of inertia (\( I \)) is given by:

• \( T = 2\pi \sqrt{\frac{I}{\kappa}} \)

Rearranging this formula allows us to solve for the moment of inertia, which is critical for applications in engineering and design, where knowing how an object will react during oscillations can inform safety and effectiveness.