Problem 40 - College Physics

Question

\(\bullet\) Two people carry a heavy electric motor by placing it on a light board 2.00 m long. One person lifts at one end with a force of \(400.0 \mathrm{N},\) and the other lifts at the opposite end with a force of 600.0 \(\mathrm{N}\) . (a) Start by making a free-body diagram of the motor. (b) What is the weight of the motor? (c) Where along the board is its center of gravity located?

Step-by-Step Solution

Answer

The weight of the motor is 1000 N, and its center of gravity is 1.2 meters from the 400 N force end.

Step 1: Identify the Forces on the Motor

In this step, consider the forces acting on the electric motor. There are two vertical forces: 400 N at one end and 600 N at the opposite end of the light board. These forces are due to the people lifting the motor.

Step 2: Draw the Free-Body Diagram

Draw a diagram with the light board on a horizontal plane. Assign designated points at each end for the lifting forces: 400 N at one end (let's call it point A) and 600 N at the other end (point B). Represent the weight of the motor, which acts downwards at its center of gravity (unknown position, x meters from point A).

Step 3: Calculate the Weight of the Motor

Since the board is in static equilibrium, the sum of the upward forces must equal the downward forces (the weight of the motor). Hence,\[ W = 400 + 600 = 1000 \text{ N} \]

Step 4: Determine the Center of Gravity

Use the moment equation about any point to find out where the center of gravity is located. Conventionally, choose point A for calculation. The total moment about point A due to the forces must be zero:\[ 600 \times 2.0 = W \times x \]Substitute the value of the weight, W:\[ 600 \times 2.0 = 1000 \times x \]Solving for x gives:\[ x = \frac{600 \times 2.0}{1000} = 1.2 \text{ m} \]

Step 5: Conclude the Positions

From the calculations, the center of gravity of the motor is found to be 1.2 meters from the end where the force is 400 N.

Key Concepts

Free-Body Diagram

A free-body diagram is a visual representation used to depict all the forces acting upon a single object. It's a crucial step in analyzing problems involving physics concepts like static equilibrium. In the case of our example, the first step involves sketching this diagram to understand and calculate the forces involved.

When creating a free-body diagram for the electric motor sitting on a board, you start by identifying all forces acting upon the motor:

• Two upward forces—one of 400 N from one person and 600 N from the other.
• Downward weight force of the motor, which is unknown until further calculations but acts at the motor's center of gravity.

Drawing this out helps visualize how forces interact and balance, which is essential for solving problems. Remember that each force is represented by an arrow pointing in the direction the force is applied, with the length representing the force's magnitude.

Center of Gravity

The center of gravity is the point at which the distribution of an object's weight is balanced. In simpler terms, it is where you could balance the object on the tip of a pencil. However, it is not always the geometric center.

For the electric motor on the board:

• The weight of the motor acts downward from its center of gravity.
• Using the concept of static equilibrium, where all torques and forces balance out, we determine that the center of gravity is 1.2 meters from the end where the 400 N force is applied.

This measured position ensures the balancing of the motor, with the forces from both individuals lifting the board perfectly distributed to maintain equilibrium.

Torque

Torque refers to the rotational effect of a force applied at a distance from a pivot point. It's pivotal in understanding how forces cause objects to rotate and is often seen in problems involving rotational equilibrium.

In the context of this exercise:

• The torque about point A, contributed by the 600 N force at the opposite end of the board, equals \( 600 \times 2.0 \; \text{m} = 1200 \; \text{Nm} \).
• The static equilibrium condition means the total torque about any point must equal zero. Therefore, the torque due to the motor's weight \( 1000 \; \text{N} \) at distance \( x \) must balance this: \( 1000 \times x \).

By calculating with the equation \( 1200 = 1000 \times x \), we find that \( x = 1.2 \; \text{m} \). Hence, understanding torque is vital in determining the position of the center of gravity and ensuring equilibrium.