Question

A child in a tree house uses a rope attached to a basket to lift a \(22-\mathrm{N}\) dog upward through a distance of \(4.7 \mathrm{~m}\) into the house. How much work does the child do in lifting the dog?

Short answer

The child does 103.4 J of work.

Step-by-step solution

Understand the Work Formula

Work is calculated using the formula \( W = F \times d \times \cos(\theta) \), where \( W \) is the work done, \( F \) is the force applied, \( d \) is the distance moved, and \( \theta \) is the angle between the force and the direction of movement. If the movement is in the direction of the force, \( \theta = 0 \), and thus \( \cos(0) = 1 \).

Identify Given Information

From the problem, we know that the force \( F \) is the weight of the dog, which is \( 22 \mathrm{~N} \), and the distance \( d \) the dog is lifted is \( 4.7 \mathrm{~m} \). The angle \( \theta \) between the force and direction of movement is \( 0 \) degrees since the force and movement are in the same direction.

Substitute into the Work Formula

The formula simplifies to \( W = F \times d \). Substitute the known values: \( F = 22 \mathrm{~N} \) and \( d = 4.7 \mathrm{~m} \).

Calculate the Work Done

Perform the multiplication to find the work: \( W = 22 \times 4.7 = 103.4 \mathrm{~J} \).

Determine the Final Answer

The work done by the child in lifting the dog is \( 103.4 \mathrm{~J} \) because the calculation showed that \( W = 103.4 \mathrm{~J} \).

Key concepts

Force

In physics, **force** is a crucial concept; it's what causes an object to change its motion. Force has both magnitude and direction, making it a vector quantity. When you lift something, like the child in the tree house lifting a dog, the force applied is often equal to the object's weight, assuming it's being moved vertically.

• The **weight of the dog** acts as the force in this scenario. It is given as 22 Newtons (N), which tells us how much pull is needed to counteract gravity and lift the dog up.
• Forces can be exerted in different directions, but in this problem, since the movement is vertical, they are aligned, evidenced by the fact that the angle involved is zero, simplifying our calculations significantly.

Understanding force in terms of weight and how it impacts movement allows us to move objects effectively and is foundational for many engineering and physics problems.

Distance

**Distance** refers to how far an object has moved, and it plays a direct role in determining work done. In this context, it's about the path length over which the force is applied.

• For the tree house scenario, the child lifts the dog a total of 4.7 meters (m). This is the vertical distance that matters for calculating work.
• The greater the distance over which a force is applied, the more work is required. However, if the movement is against force (e.g., against gravity), more energy or work is needed.

Distance doesn't carry direction, unlike displacement in physics, which is why it's considered a scalar quantity. Here, it helps multiply with force to determine the work, assuming constant force and no special angles involved.

Angle

The concept of **angle** is integral when analyzing work in physics because it dictates how effectively a force contributes to motion. In mathematical terms, this is captured by the cosine of the angle between the force direction and the direction of movement.

• In our problem, the angle \( \theta \) is 0 degrees, meaning the force is applied directly along the line of movement, vertically.
• When the angle is 0, \( \cos(0) = 1 \), which means the entire force contributes to doing work on the object being moved.
• If the angle were different, such as if the rope was pulling the basket at an angle, the effective force would be \( F \times \cos(\theta) \), reducing the work done because only part of the force would be moving the object in the desired direction.

Knowing how angles affect force application is essential in fields like engineering, where optimizing force to achieve maximum effectiveness is often critical.