Question

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

Short answer

The child does 103.4 Joules of work.

Step-by-step solution

Understand the Concept of Work

Work is defined as the amount of energy transferred by a force acting through a distance. It can be calculated using the formula: \[ \text{Work} = \text{Force} \times \text{Distance} \times \cos(\theta) \] where \( \theta \) is the angle between the force and the direction of the movement.

Identify the Values Provided

In this problem, the force is the weight of the dog which is 22 N, the distance the dog is lifted is 4.7 m, and the motion of the dog is in the direction of the force applied (vertically upward), so \( \theta = 0^\circ \).

Apply the Formula for Work

Substitute the values into the work formula. Since the angle \( \theta = 0^\circ \), the cosine of this angle is 1. Thus, the formula simplifies to:\[ \text{Work} = 22 \, \text{N} \times 4.7 \, \text{m} \times \cos(0^\circ) = 22 \, \text{N} \times 4.7 \, \text{m} \times 1 \]

Calculate the Work Done

Perform the multiplication to find the work done: \[ \text{Work} = 22 \, \text{N} \times 4.7 \, \text{m} = 103.4 \, \text{J} \]Thus, the work done by the child in lifting the dog is 103.4 Joules.

Key concepts

Work calculation

When we talk about work in physics, we're referring to the amount of energy transferred when a force moves an object over a distance. The formula for calculating work is quite straightforward:

• \[ \text{Work} = \text{Force} \times \text{Distance} \times \cos(\theta) \]

Here:

• "Work" is measured in Joules (J).
• The "force" is in Newtons (N).
• "Distance" is how far the object moves in meters (m).
• \( \theta \) is the angle between the force and the direction of movement.

In our exercise, since the movement is vertical, \( \theta = 0^\circ \) which simplifies the cosine part of the formula to 1. So, calculating work becomes a matter of multiplying force by distance. If the direction of force and movement were different, you'd need to adjust using the cosine of the angle between them.

Physics problem-solving

Solving physics problems might feel daunting at first, but breaking them down into simple steps makes understanding much easier. When approaching any physics problem, follow these guidelines:

• Identify what's known: Collect all the given values from the problem statement. For instance, in this problem, we know the force is 22 N, and the distance is 4.7 m.
• Identify what's unknown: Clearly define what you need to find. Here, it's the amount of work done.
• Use relevant formulas: Choose the formula that relates the known values to what's unknown. Often, this is based on fundamental principles (like the work formula).
• Substitute values and solve: Insert the known quantities into the equation and perform any necessary calculations.

Remember, practice is key in physics. The more you practice problem-solving, the more intuitive it becomes. So, take your time working through each step carefully to build that familiarity.

Force and distance

The concepts of force and distance play crucial roles in understanding work and energy. Here is what they involve: **Force** - Force is essentially a push or pull acting on an object, measured in Newtons (N). - It can cause an object to start moving, stop, or change direction. - In this scenario, the force is the weight of the dog, equivalent to its gravitational force (22 N). **Distance** - Distance refers to how far an object moves, measured in meters (m). - It’s the path taken by the object due to the force applied. Here, the dog moves vertically up for 4.7 meters.
Understanding how force over a distance contributes to work is crucial. Work is only done when displacement occurs in the direction of the force. If the dog were simply hanging stationary, even with a force applied, no work would be done since there's no movement. Thus, both the magnitude of the force and the distance it moves an object are essential for calculating work.