Problem 46 - College Physics

Question

A woman is standing in an elevator holding her 2.5 -kg briefcase by its handles. Draw a free-body diagram for the briefcase if the elevator is accelerating downward at 1.50 \(\mathrm{m} / \mathrm{s}^{2}\) and calculate the downward pull of the briefcase on the woman's arm while the elevator is accelerating.

Step-by-Step Solution

Answer

The downward pull of the briefcase is 28.25 N.

Step 1: Identify Forces Acting on the Briefcase

First, identify all the forces acting on the briefcase. The forces include the gravitational force (weight) acting downward and the tension in the handle acting upward, which is equivalent to the force exerted by the woman's hand.

Step 2: Draw the Free-Body Diagram

Draw the free-body diagram of the briefcase. Represent the forces: the weight of the briefcase as a downward arrow labeled as \( F_g = mg \), where \( m = 2.5 \, \text{kg} \) and the acceleration due to gravity \( g=9.8 \, \text{m/s}^2 \). The tension \( T \) in the handle acts upward.

Step 3: Write the Equation of Motion

Since the elevator is accelerating downward, apply Newton's second law: \[ \sum F = ma \]. Here, the net force is the weight minus the tension. Plug in acceleration, \( a = -1.50 \, \text{m/s}^2 \), to get \( mg - T = ma \).

Step 4: Solve for Tension

Substitute known values into the equation from Step 3: \( 2.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 - T = 2.5 \, \text{kg} \times (-1.50 \, \text{m/s}^2) \). Simplify and solve for \( T \).

Step 5: Perform the Calculation

Calculate the terms: \( 2.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 24.5 \, \text{N} \) and \( 2.5 \, \text{kg} \times (-1.50 \, \text{m/s}^2) = -3.75 \, \text{N} \). Thus, \( 24.5 \, \text{N} - T = -3.75 \, \text{N} \). Solve for \( T \): \( T = 24.5 \, \text{N} + 3.75 \, \text{N} = 28.25 \, \text{N} \).

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that explains how the motion of an object is affected by forces. In simple terms, it states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. This can be expressed with the equation:

• \( F = ma \)

Where:

• \( F \) is the force applied
• \( m \) is the mass of the object
• \( a \) is the acceleration of the object

This principle helps us understand scenarios where multiple forces are acting on a single body, just like our briefcase in the elevator.
By substituting the forces into Newton's second law, we can solve for unknowns such as tension or acceleration.
In our case, we calculated the tension in the handle of the briefcase while it was under the influence of both gravitational force and the downward acceleration of the elevator.

Gravitational Force

Gravitational force is a force that attracts two bodies towards each other, and it's what gives objects their weight.
It is fundamentally linked to the mass of the bodies and the distance between them, but near the Earth's surface, it is simplified as the weight of the object.

• The formula for calculating weight is \( F_g = mg \)

Where:

• \( F_g \) is the gravitational force (weight)
• \( m \) is the mass of the object
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \) on Earth)

In our elevation scenario, the gravitational force acts downward. The briefcase's weight is calculated by multiplying its mass (2.5 kg) by the gravitational acceleration, which gives us a force of 24.5 Newtons acting downwards.
Understanding gravitational force is crucial when analyzing motion and the forces exerted on objects.

Tension

Tension is a force exerted by a string, rope, cable, or in this case, the handle of a briefcase, when it is pulled tight by forces acting from opposite ends.
In the context of our problem, tension is the force exerted by the woman's hand to hold the briefcase while it experiences downward acceleration.
To solve for tension, we used the rearranged formula from Newton’s second law:

• \( T = mg - ma \)

Plugging in the values calculated in the previous steps:

• \( mg = 24.5 \, \mathrm{N} \) (gravitational force)
• \( ma = -3.75 \, \mathrm{N} \) (force due to elevator's acceleration)

We found that the tension in the handle was 28.25 Newtons, calculated by adding the absolute value of both forces since they act in opposite directions.
Understanding tension is vital when solving problems of objects suspended or held with strings or handles in various scenarios.