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. When the elevator has an upward acceleration of magnitude $2.50$ m/s², what does the scale read?

Verified step by step guidance

1. Step 1: Begin by identifying the forces acting on you while standing on the scale in the elevator. These forces include your weight (gravitational force, Fg = mg) acting downward and the normal force (Fn) exerted by the scale acting upward. The scale reading corresponds to the magnitude of the normal force.
2. Step 2: Write Newton's second law for the vertical direction: \( F_{net} = ma \). Here, \( F_{net} \) is the net force acting on you, \( m \) is your mass, and \( a \) is the acceleration of the elevator. The net force is given by \( F_{net} = F_n - F_g \), where \( F_n \) is the normal force and \( F_g \) is your weight.
3. Step 3: Calculate your mass using your normal weight. Recall that \( F_g = mg \), where \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)). Rearrange to find \( m = \frac{F_g}{g} \). Substitute \( F_g = 625 \, \text{N} \) and \( g = 9.8 \, \text{m/s}^2 \) to determine your mass.
4. Step 4: Substitute the values of \( m \), \( g \), and \( a \) into the equation \( F_n = F_g + ma \). Here, \( F_g \) is your weight, \( m \) is your mass, and \( a \) is the upward acceleration of the elevator (\( 2.50 \, \text{m/s}^2 \)). This equation accounts for the additional force required to accelerate you upward.
5. Step 5: Solve for \( F_n \), which represents the scale reading. The result will be the sum of your normal weight and the additional force due to the elevator's upward acceleration. Ensure all units are consistent during substitution and calculation.

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation used to visualize the forces acting on an object. In this case, it would illustrate the forces on the person in the elevator, including gravitational force (weight) acting downward and the normal force exerted by the scale acting upward. This helps in analyzing the net force and acceleration of the object.

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. In the context of the elevator, this law helps determine the net force when the elevator accelerates upward, allowing us to calculate the apparent weight shown on the scale.

Apparent Weight

Apparent weight is the normal force experienced by an object in a non-inertial frame, such as an accelerating elevator. It differs from true weight due to the effects of acceleration. When the elevator accelerates upward, the apparent weight increases, which is what the scale measures, reflecting the combined effect of gravitational force and the additional force due to the elevator's acceleration.