You walk into an elevator, step onto a scale, and push the 'up' button. You recall that your normal weight is $625$ N. Draw a free-body diagram. If you hold a $3.85$-kg package by a light vertical string, what will be the tension in this string when the elevator accelerates as in part (a)? Note: Part (a) asked what does the scale read when the elevator has an upward acceleration of magnitude $2.50$ m/s².

Verified step by step guidance

1. Step 1: Begin by analyzing the forces acting on you in the elevator. Your normal weight is 625 N, which is the gravitational force acting downward. When the elevator accelerates upward, there is an additional force due to the acceleration. Draw a free-body diagram showing two forces: the gravitational force (weight) acting downward and the normal force exerted by the scale acting upward.
2. Step 2: Use Newton's second law of motion, \( F_{net} = ma \), to calculate the net force acting on you. The net force is the difference between the upward normal force (scale reading) and the downward gravitational force. Express this as \( F_{net} = F_{normal} - F_{gravity} \). Rearrange to find \( F_{normal} = F_{gravity} + ma \).
3. Step 3: For the package, consider the forces acting on it. The package has a gravitational force \( F_{gravity, package} = m_{package} g \) acting downward and a tension force \( T \) in the string acting upward. When the elevator accelerates upward, the net force on the package is \( F_{net, package} = m_{package} a \). Use Newton's second law to express the tension in the string as \( T = F_{gravity, package} + m_{package} a \).
4. Step 4: Substitute the known values into the equations. For the package, \( m_{package} = 3.85 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \). Calculate \( F_{gravity, package} = m_{package} g \). Then, use the acceleration \( a \) from part (a) to find the tension \( T \) in the string.
5. Step 5: Ensure all units are consistent and verify your calculations. The tension in the string depends on the elevator's acceleration \( a \), which you determined in part (a). Once \( a \) is known, substitute it into the equation for \( T \) to find the tension in the string.

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation used to visualize the forces acting on an object. In this context, it helps identify the forces on the person in the elevator, including gravitational force (weight) and the normal force from the scale. This diagram is essential for analyzing the net force and understanding how the forces interact during the elevator's acceleration.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, expressed as F = ma. This principle is crucial for determining the tension in the string when the elevator accelerates, as it allows us to calculate the net force acting on the package and how it relates to the forces of tension and gravity.

Tension in a String

Tension is the force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends. In this scenario, the tension in the string holding the package will vary depending on the elevator's acceleration. Understanding how to calculate tension involves considering both the weight of the package and the additional force due to the elevator's acceleration.