Answer:
The radius of gyration (k) of a pendulum is a measure of how the mass is distributed around the axis of rotation. It is defined as the square root of the ratio of the moment of inertia (I) of the pendulum about the given axis to its total mass (m). Mathematically, it is expressed as:
k = √(I / m)
To determine the radius of gyration of the given pendulum, we need to calculate its moment of inertia about the given axis and divide it by the total mass of the pendulum.
Given:
Mass of the circular plate (m1) = 7 kg
Mass of the slender rod (m2) = 3 kg
The moment of inertia of a circular plate rotating about an axis perpendicular to its plane passing through its center (I1) is given by the formula:
I1 = (1/2) * m1 * r1^2
where r1 is the radius of the circular plate.
The moment of inertia of a slender rod rotating about an axis perpendicular to its length passing through its center (I2) is given by the formula:
I2 = (1/3) * m2 * L^2
where L is the length of the slender rod.
Since the pendulum consists of both the circular plate and the slender rod, the total moment of inertia of the pendulum about the given axis (I) is the sum of I1 and I2:
I = I1 + I2
Plugging in the given values:
I1 = (1/2) * 7 * r1^2
I2 = (1/3) * 3 * L^2
We would need to know the values of r1 and L in order to calculate the moment of inertia I and subsequently the radius of gyration k. If you provide the values for r1 and L, I would be happy to help you calculate the radius of gyration of the pendulum about the given axis passing through point O.
Step-by-step explanation: