A rigid bar (negligible mass) of length 80 cm connects a large sphere with a mass (m₁) of 25 g to a small sphere with an unknown mass (m₂). The large sphere is located at one end of the bar, with the center of mass of the bar located 22 cm away from this sphere. Determine the mass of the sphere (in grams) at the other end of the bar.

Verified step by step guidance

1. Identify the given values: the length of the bar is 80 cm, the mass of the large sphere (m_1) is 25 g, and the center of mass is 22 cm from the large sphere.
2. Let the mass of the small sphere be m_2 (in grams).
3. Use the formula for the center of mass of a system of particles: \( \frac{m_1 \cdot d_1 + m_2 \cdot d_2}{m_1 + m_2} = \text{center of mass position} \), where \( d_1 \) and \( d_2 \) are the distances of m_1 and m_2 from the reference point.
4. Set the reference point at the large sphere, so \( d_1 = 0 \) and \( d_2 = 80 \) cm. The center of mass is 22 cm from the large sphere, so substitute these values into the equation: \( \frac{25 \cdot 0 + m_2 \cdot 80}{25 + m_2} = 22 \).
5. Solve the equation for m_2 by multiplying both sides by \( 25 + m_2 \), then isolate m_2 on one side of the equation.

Key Concepts

Center of Mass

The center of mass is a point that represents the average position of the mass distribution of an object or system. For a rigid body, it can be calculated by taking the weighted average of the positions of all the masses involved. In this problem, the center of mass of the bar is crucial for determining the balance point between the two spheres, allowing us to set up an equation based on their distances and masses.

Equilibrium Condition

In physics, an object is in equilibrium when the sum of the forces and the sum of the torques acting on it are zero. For this problem, the rigid bar will be in rotational equilibrium, meaning that the moments (torques) created by the weights of the two spheres about the center of mass must balance each other. This condition allows us to derive a relationship between the masses of the two spheres.

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (in this case, the center of mass). In this scenario, the torques generated by the weights of the large and small spheres about the center of mass must be equal for the system to be in equilibrium. Understanding how to calculate and set these torques equal is essential for solving for the unknown mass.