Problem 21 - Applied Physics

Question

A delivery person carries a \(215-\mathrm{N}\) box up stairs \(4.20 \mathrm{~m}\) vertically and \(6.80 \mathrm{~m}\) horizontally. (a) How much work does the delivery person do? (b) How much work does the delivery person do in carrying the box down the stairs?

Step-by-Step Solution

Answer

(a) 903 J; (b) 0 J

Step 1: Understand the Problem

We need to determine two specific amounts of work done by the delivery person. First, the work done while carrying the box up the stairs and then the work done while carrying the box down the stairs. We'll use the concept that work is done when a force causes displacement in the direction of the force.

Step 2: Calculate Vertical Work Done

The work done while moving vertically is due to the gravitational force. The formula for work is given by \( W = F \cdot d \cdot \cos(\theta) \), where \( F \) is the force, \( d \) is the distance moved in the direction of the force, and \( \theta \) is the angle between the force and the direction of the movement. Here, the box is moved vertically upwards against gravity (whose force is 215 N), over a distance of 4.20 m, with \( \theta = 0 \) degrees: \( W = 215 \cdot 4.20 \cdot \cos(0) = 903 \, \text{J} \).

Step 3: Calculate Horizontal Work Done

For horizontal movement, the displacement is parallel, but perpendicular to the gravitational force. Thus, \( \theta = 90 \) degrees and \( \cos(90) = 0 \). Hence, the work done in the horizontal direction is \( W = 215 \cdot 6.80 \cdot \cos(90) = 0 \, \text{J} \).

Step 4: Compute Total Work Done Upwards

Since the work done horizontally is zero because there is no resistance, only the vertical work counts. Therefore, the total work done in carrying the box upwards is the work calculated from the vertical movement which is 903 J.

Step 5: Detemine Work Done Downwards

When moving the box down the stairs, the gravitational force is aiding the movement. Thus, the delivery person does no work, as the gravitational force is in the same direction as the movement and does the work itself. Therefore, the work done by the delivery person is 0 J.

Key Concepts

Gravitational Force

In physics, gravitational force is a fundamental force that acts between two masses. On Earth, it is the force that gives objects weight and pulls them towards the center of the planet. For example, when the delivery person carries a 215-Newton box, this force is essentially the object's weight acting downwards.
Gravitational force plays a critical role when performing work. When lifting an item vertically against gravity, work is done to overcome this force. Understanding gravitational force helps us calculate how much energy is needed to move objects vertically. Remember, the gravitational force on a box is constant as long as we are close to Earth, making the calculations more straightforward.

Vertical Displacement

Vertical displacement refers to the movement of an object along the vertical axis, or up and down. In our context, the delivery person is moving the box upwards, effectively achieving vertical displacement. The vertical distance moved here is 4.20 meters.
When we lift an object vertically, we apply a force to overcome gravity. The work done is significant because this force is in the same direction as the displacement. This vertical movement is essential in the calculation of work since the force of gravity has to be overcome by the delivery person. Remember, the work done is the product of force, the distance moved vertically, and the cosine of the angle between force and movement.

Horizontal Displacement

Horizontal displacement is the movement of an object from one point to another along a horizontal plane. In the case of the exercise, the box was moved 6.80 meters horizontally. However, while horizontal movement might seem significant, it plays an entirely different role in the calculation of work with gravity.
Gravitational force acts vertically, so when an object moves horizontally, there's no vertical force to resist or aid this displacement. This makes the angle between the gravitational force and horizontal displacement 90 degrees. Mathematically, since the cosine of 90 degrees is zero, the work done moving horizontally under a gravitational context is zero.

Work Done Calculation

Calculating work involves understanding three crucial components: force, displacement, and the angle between them. The formula used for calculating work is:

• Where:
  • \( W = F \cdot d \cdot \cos(\theta) \)
  • \( W \) is work done
  • \( F \) is the applied force
  • \( d \) is the displacement
  • \( \theta \) is the angle between force and displacement direction

In our exercise, for moving the box upward, the angle \( \theta \) is 0 degrees because the direction of force and direction of displacement align. So, \( \cos(0) = 1 \), simplifying the work done calculation as just force times distance. For horizontal displacement, \( \theta \) is 90 degrees, meaning no work is done since \( \cos(90) = 0 \). This difference highlights why understanding the context of each movement is essential in physics.

Direction of Force

The direction of a force significantly influences how much work is done. In physics, force is a vector, meaning it has both magnitude and direction. When calculating work, the direction of the force determines how much work is expended or transferred to an object.
In the vertical direction of our exercise, the force needed to move the box is aligned with the gravitational pull, opposing it directly when moving upward. Here, work is done against gravity. When moving an object horizontally, and perpendicular to gravity, no work is required in relation to gravitational force. Understanding the angle between force direction and displacement is essential, as it directly affects work done in each movement scenario.