Problem 38 - College Physics

Question

Two people are carrying a uniform wooden board that is 3.00 \(\mathrm{m}\) long and weighs 160 \(\mathrm{N}\) . If one person applies an upward force equal to 60 \(\mathrm{N}\) at one end, at what point and with what force does the other person lift? Start with a free-body diagram of the board.

Step-by-Step Solution

Answer

The other person lifts with 100 N at 2.4 m from the first person.

Step 1: Analyze the Problem

The board is uniform, 3.00 m long, and weighs 160 N. One person applies an upward force of 60 N at one end. Our goal is to find the force and position where the other person must lift to balance the torques.

Step 2: Draw the Free-Body Diagram

Represent the board as a horizontal beam. Mark the forces acting: 60 N upward at one end, the weight of the board (160 N) acting at the center (1.5 m from both ends), and an unknown force F acting at an unknown position x (from the first person).

Step 3: Use Torque Equilibrium

Set up the torque equilibrium about the pivot point (where the first person lifts):\[ 60 imes 0 + F imes x - 160 imes 1.5 = 0 \]

Step 4: Solve for F and x

Since we want equilibrium, the sum of forces is zero:- For forces: \[ F + 60 = 160 \] Thus, \[ F = 100 \ N \]- Substituting F into torque equation: \[ 100x - 240 = 0 \] \[ 100x = 240 \] \[ x = 2.4 \ m \]

Step 5: Conclusion

The other person lifts with a force of 100 N, positioned 2.4 m from the first person (or 0.6 m from the opposite end).

Key Concepts

Free-Body Diagram

A Free-Body Diagram (FBD) is a simple graphical illustration used in physics to visualize the forces acting on an object or system. This method is crucial for understanding and solving problems involving forces and torques. To draw a proper FBD:

• Identify the object of interest, in this case, the wooden board.
• Represent the object as a simple shape, like a horizontal line for the board.
• Mark and label all forces acting on the object. In our problem, these are the 60 N upward force, the board's weight of 160 N centered at 1.5 m from any end, and the unknown force to be determined.

The Free-Body Diagram simplifies the complexity of real-world problems and is vital in organizing information clearly. With this visual representation, it becomes easier to apply the laws of physics to calculate the unknown values. This is why starting with an FBD is a fundamental step in physics problem-solving.

Force Analysis

Force Analysis in physics involves identifying and calculating all forces acting upon an object. It's a critical step in understanding how an object interacts with its environment and is imperative in Torque Equilibrium calculations.
When performing force analysis, consider the following steps:

• Ensure all forces are identified: gravitational, applied, normal, and any other relevant forces.
• Calculate any known forces. For our board, we know there's a 60 N force applied upwards and the board's weight, 160 N, acting downward.
• Determine the unknown forces using equations. To find the point of equilibrium, ensure that the total sum of forces equals zero.

In this exercise, we want the board to remain stable. This means the upward and downward forces must balance, setting the stage for the Torque Equilibrium calculations. The analysis eventually shows the other person must exert a force of 100 N to maintain equilibrium.

Physics Problem Solving

Physics Problem Solving capitalizes on systematic approaches to unravel questions posed by physical situations. It often requires a blend of intuition, analytical skills, and methodical solutions due to the multifaceted nature of physics problems.
In tackling problems like this one, several steps are key:

• Begin with comprehensive problem analysis. Understand what is being asked and what is given.
• Employ models like the Free-Body Diagram to visualize the setup.
• Use physics principles such as the equilibrium principle, stating that for a system at rest, the sum of forces and torques must be zero.
• Mathematically solve for unknowns through algebraic manipulation.
• Cross-verify solutions to ensure they make physical sense in the context of the problem.

For our wooden board problem, clear problem-solving techniques helped us determine that to achieve torque equilibrium, the second person should lift with 100 N at a position 2.4 m from the first lifter. These techniques help in approaching not just simple balance problems but a wide array of physics challenges effectively.