A physical pendulum consists of a meter stick that is pivoted at a small hole drilled through the stick a distance x from the 50.0-cm mark. The period of oscillation is observed to be 2.50s. Find the distance x.

To find the distance x from the 50.0-cm mark for a physical pendulum, we can use the formula for the period of a physical pendulum, which is given by:

Formula for the Period of a Physical Pendulum

The period (T) of a physical pendulum can be calculated using the formula:

T = 2π√(I/mgh)

Where:

• T = period of oscillation
• I = moment of inertia of the pendulum about the pivot point
• m = mass of the pendulum
• g = acceleration due to gravity (approximately 9.81 m/s²)
• h = distance from the pivot point to the center of mass

Step 1: Identifying Parameters

In this case, we have a meter stick (which is 1 meter long) pivoted at a distance x from the 50.0-cm mark. The total length of the stick means the distances to consider are:

• Length of the stick (L) = 1 m
• Mass (m) = uniform, so we can assume it to be some constant value (we can cancel it out later)
• Distance to center of mass (h) = (1/2)L = 0.5 m from one end

Step 2: Calculate Moment of Inertia (I)

The moment of inertia for a uniform rod about an axis through one end is given by:

I = (1/3)mL²

However, since our pivot is at a distance x from the center, we need to adjust this using the parallel axis theorem:

I = (1/3)mL² + m d²

Where d = distance from the pivot to the center of mass = (50 cm - x) = (0.5 - x) m. So:

I = (1/3)m(1)² + m(0.5 - x)²

I = (1/3)m + m(0.5 - x)²

Step 3: Putting it All Together

Now substitute I into the period formula:

T = 2π√((1/3)m + m(0.5 - x)²) / (mg(0.5 - x))

Cancel out mass (m) from the numerator and denominator:

T = 2π√((1/3) + (0.5 - x)²) / (g(0.5 - x))

Step 4: Solve for x

We know that T = 2.50 s. Plugging in the values, we get:

2.50 = 2π√((1/3) + (0.5 - x)²) / (9.81(0.5 - x))

Squaring both sides will help eliminate the square root:

(2.50)² = (2π)²((1/3) + (0.5 - x)²) / (9.81²(0.5 - x)²)

Now simplifying will lead to an equation where you can isolate x. This will involve some algebra, but you will find that there will be two possible solutions for x within the bounds of the stick's length.

Final Thoughts

This approach will allow you to derive the specific distance x that satisfies the condition of the observed period of oscillation of the pendulum. Keep in mind that the system's physical constraints (like the length of the stick) will guide the possible values for x. If you go through the calculations carefully, you will arrive at the precise value for x, which will help you understand how the position of the pivot affects the oscillation period of a physical pendulum.