Problem 43 - College Physics

Question

You walk into an elevator, step onto a scale, and push the "up" button. You also recall that your normal weight is 625 \(\mathrm{N}\) . Start each of the following parts with a free-body diagram. (a) If the elevator has an acceleration of magnitude \(2.50 \mathrm{m} / \mathrm{s}^{2},\) what does the scale read? (b) If you start holding a 3.85 kg package by a light vertical string, what will be the tension in this string once the elevator begins accelerating?

Step-by-Step Solution

Answer

(a) The scale reads 784 N. (b) The tension in the string is 47.4 N.

Step 1: Analyze Forces on the Scale (Part a)

Firstly, draw a free-body diagram for yourself inside the elevator. There are two main forces acting on you: the gravitational force (weight) acting downwards, and the normal force exerted by the scale upwards. The net force causes the acceleration of the elevator. The gravitational force can be calculated as the normal weight, which is given as 625 N.

Step 2: Calculate the Effective Force for Scale Reading (Part a)

The net force (F_{net}) acting on you is the difference between the normal force (N) and your weight (mg), wheremg = 625 \, \text{N}. Using Newton's second law,F_{net} = ma, wheremis your mass andais the acceleration of the elevator (2.50 m/s²).\[ma = N - mg\]Rearrange to findN:\[N = ma + mg\]

Step 3: Solve for Normal Force (Scale Reading) (Part a)

To solve for the normal force (N), first find your mass (m) using your weight:\[m = \frac{625}{9.81} \approx 63.7 \, \text{kg}\]Now substitute the known values into the equation:\[N = (63.7 \times 2.50) + 625\]\[N \approx 159 + 625 = 784 \, \text{N}\]Thus, the scale will read approximately 784 N.

Step 4: Analyze Forces on the Package (Part b)

Draw a free-body diagram for the package you’re holding. The forces acting on the package are similar to yours: gravitational force ( mg ) downwards and tension in the string ( T ) upwards. The upward acceleration of the elevator affects the tension in the string.

Step 5: Use Newton's Second Law for the Package (Part b)

Using Newton's second law for the package, the net force is:\[T - mg = ma\]wheremis the mass of the package (3.85 kg) andais the acceleration of the elevator. Rearrange to findT:\[T = m(g + a)\]

Step 6: Solve for Tension in the String (Part b)

Substitute the known values into the equation fora = 2.50 \, \mathrm{m/s^2} andm = 3.85 \, \text{kg}:\[T = 3.85(9.81 + 2.50)\]\[T = 3.85 \times 12.31\]\[T \approx 47.4 \, \text{N}\]Thus, the tension in the string would be approximately 47.4 N.

Key Concepts

Free-body diagram

A free-body diagram is a simple drawing used to visualize the forces acting on an object. To create one, you draw the object and represent each force as an arrow pointing in the direction the force is acting. The length of the arrow reflects the magnitude of the force.

For the scenario in the elevator, you need two free-body diagrams: one for yourself and one for the package you hold.

• **For yourself**: The forces are your gravitational weight pointing downwards and the normal force from the scale pointing upwards.
• **For the package**: The forces are the gravitational force of the package pointing downwards and the tension in the string pointing upwards.

These diagrams help in understanding what forces are acting and in which direction they are applied. They are crucial in the application of Newton's Second Law, as they let you visually identify the net force and its impact on the object’s motion.

Weight and normal force

The concept of weight and normal force is vital when you're inside the elevator. Your weight is the gravitational force acting on you, calculated as your mass multiplied by the acceleration due to gravity, which is approximately 9.81 m/s². In the problem, your weight is 625 N.

The normal force, on the other hand, is the force exerted by the scale which acts upwards. It is this force that the scale measures to show your apparent weight. If the elevator accelerates upwards, the normal force increases compared to the gravitational force because it has to counter your weight and provide the extra force for acceleration.

Using Newton's Second Law, we find out the normal force with the formula: \[ N = ma + mg \]Where:

• \( m \) is your mass.
• \( a \) is the elevator's acceleration.
• \( g \) is the acceleration due to gravity.

Therefore, the scale reads higher than your actual weight (625 N) when the elevator is accelerating upwards.

Tension in a string

Tension in a string is another significant concept when holding a package inside the accelerating elevator. As the elevator ascends, the string must not only support the weight of the package but also provide an upward force to accelerate it.

Like the normal force, tension also changes with acceleration. To find the tension, use the Newton's Second Law, rearranged for this context:\[ T = m(g + a) \]Where:

• \( T \) is the tension in the string.
• \( m \) is the mass of the package.
• \( a \) is the elevator's upward acceleration.

This formula calculates how much force the string needs to exert to overcome the gravitational pull and to provide acceleration to the package. Hence, the tension in the string turns out to be more than simply the gravitational weight of the package, reflecting the additional effort needed to move upwards with the elevator.