A 4.0-m-long, 500 kg steel beam extends horizontally from the point where it has been bolted to the framework of a new building under construction. A 70 kg construction worker stands at the far end of the beam. What is the magnitude of the torque about the bolt due to the worker and the weight of the beam?

Verified step by step guidance

1. Step 1: Understand the concept of torque. Torque is the rotational equivalent of force and is calculated using the formula: $\tau =r\cdot F\cdot \sin \left(\theta \right)$, where $r$ is the distance from the pivot point, $F$ is the force applied, and $\theta$ is the angle between the force and the lever arm. In this problem, the angle is 90 degrees, so $\sin \left(90\right)=1$.
2. Step 2: Calculate the torque due to the construction worker. The force exerted by the worker is his weight, which is given by $F=m\cdot g$, where $m$ is the mass of the worker (70 kg) and $g$ is the acceleration due to gravity (approximately 9.8 m/s²). The distance $r$ is the length of the beam (4.0 m). Use the formula for torque: $\tau =r\cdot F$.
3. Step 3: Calculate the torque due to the weight of the beam. The beam's weight acts at its center of gravity, which is located at the midpoint of the beam (2.0 m from the bolt). The force exerted by the beam is its weight, given by $F=m\cdot g$, where $m$ is the mass of the beam (500 kg) and $g$ is the acceleration due to gravity (9.8 m/s²). Use the formula for torque: $\tau =r\cdot F$.
4. Step 4: Add the torques together. Since both the worker and the beam exert forces that cause rotation in the same direction about the bolt, their torques can be added directly. The total torque is the sum of the torque due to the worker and the torque due to the beam.
5. Step 5: Ensure units are consistent. Verify that all distances are in meters, masses are in kilograms, and forces are in newtons. This ensures the torque is calculated in newton-meters (N·m).

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object around a pivot point. It is calculated as the product of the force applied and the distance from the pivot point to the line of action of the force, expressed as τ = r × F, where τ is torque, r is the distance, and F is the force. In this scenario, both the weight of the beam and the worker's weight create torques about the bolt.

Center of Mass

The center of mass is the point at which the mass of an object is concentrated and can be considered to act. For uniform objects like the steel beam, the center of mass is located at its midpoint. Understanding the center of mass is crucial for calculating the torque due to the beam's weight, as it determines the effective distance from the pivot point to where the weight acts.

Equilibrium

Equilibrium in physics refers to a state where the sum of forces and the sum of torques acting on an object are zero, resulting in no net motion. In this problem, analyzing the torques created by the worker and the beam's weight about the bolt helps determine if the system is in equilibrium and allows for the calculation of the total torque acting on the beam.