Question

Calculate the work done in the following situations. A stroller is pushed \(20 \mathrm{m}\) with a constant force of \(10 \mathrm{N}\) at an angle of \(15^{\circ}\) below the horizontal.

Short answer

Question: Calculate the work done in pushing a stroller at an angle of \(15^{\circ}\) below the horizontal over a distance of \(20\mathrm{m}\) with a constant force of \(10\mathrm{N}\). Answer: The work done in pushing the stroller is approximately \(193.2 \mathrm{J}\).

Step-by-step solution

Understand the problem

We are given the following parameters: - Distance (\(d\)) = \(20 \mathrm{m}\), - Force (\(F\)) = \(10 \mathrm{N}\), - Angle below the horizontal (\(\theta\)) = \(15^{\circ}\). We need to find the work done in pushing the stroller.

Resolve the force into its horizontal component

Since the force is acting at an angle below the horizontal, we will first find the horizontal component (\(F_h\)) of the force by using the cosine of the angle: $$F_h = F \cos(\theta)$$ Plugging in the given values, we have: $$F_h = 10 \mathrm{N} \cos(15^{\circ})$$

Calculate the cosine of the angle

Now calculate the cosine of the angle \(15^{\circ}\): $$\cos(15^{\circ}) \approx 0.966$$

Calculate the horizontal component of the force

Using the cosine value, we can find the horizontal component of the force: $$F_h = 10 \mathrm{N} \times 0.966 \approx 9.66 \mathrm{N}$$

Calculate the work done

Finally, we will use the definition of work done to calculate the work done in pushing the stroller: $$W = F_h \times d$$ Plugging in the values: $$W = 9.66 \mathrm{N} \times 20 \mathrm{m} \approx 193.2 \mathrm{J}$$ The work done in pushing the stroller is approximately \(193.2 \mathrm{J}\).

Key concepts

Force Resolution

Force resolution is a fundamental concept in physics. It's used to break down a single force acting on an object into two components, typically along the horizontal and vertical axes. In the problem of the stroller, we have a force being applied at an angle to the horizontal, which means we need to resolve it to understand how much of that force is contributing to motion along the horizontal path. To resolve a force, we typically use trigonometric functions such as sine and cosine, depending on the angle given. These functions help to project the total force into components that are easier to work with in calculations. This is particularly useful when the force isn't perpendicular or parallel to the direction of movement. Force resolution allows us to:

• Identify how much of a force is pushing an object forward along the desired path.
• Understand how much of a force is working against inertia or friction.

So, when dealing with angled forces, remember: resolving into components is your way to simplify and visualize the forces at play.

Horizontal Component

In many physics problems, knowing the horizontal component of a force is crucial for calculating work done. This is because only the force parallel to the direction of movement does work. The horizontal component, often denoted as \( F_h \), is the part of the total force that acts in the horizontal direction. In our stroller example, to get this component, we use the cosine of the angle given, as it represents the adjacent side when considering a right triangle made by the force.The formula used is: \[ F_h = F \cos(\theta) \]where:

• \( F \) is the total force, and
• \( \theta \) is the angle of the force with the horizontal.

By calculating the horizontal component, we effectively determine how much of the pushing force actually contributes to moving the stroller forward. This understanding not only helps in calculating work but is also crucial in engineering to ensure forces are applied efficiently.

Cosine Function

The cosine function is a key player in calculations involving angles and forces. In physics, it helps find the horizontal component of a force when that force is applied at an angle. The cosine of an angle in a right triangle represents the ratio of the length of the adjacent side to the length of the hypotenuse. Put simply, when you have a force vector and an angle, the cosine function helps you determine how much of that force is pulling in the straight direction of the path you are interested in (in this case, along the horizontal). For familiarity:

• \( \cos(0^{\circ}) = 1 \): The whole force is horizontal.
• \( \cos(90^{\circ}) = 0 \): No horizontal component is present.
• \( \cos(15^{\circ}) \approx 0.966 \): Most of the force is acting horizontally.

Using the cosine function as we've done helps to simplify complex force interactions into calculable numbers that tell us exactly how forces are acting on an object.

Physics Problem Solving

Successfully solving a physics problem often involves breaking down complex ideas into manageable parts. This process is evident when calculating work done on an object like the stroller being pushed. The first step is understanding what is being asked: calculating work, which involves both force and distance. Next, you need to know the direction of the force and the angle at which it is applied. Once you have these details, solve the problem using logical steps:

• Step 1: Resolve the force using trigonometry to find components.
• Step 2: Use the components to calculate work done.

In our specific example, after resolving into horizontal components using cosine, we find out how much force contributes to moving the stroller. The calculated horizontal force multiplied by the distance gives the work done. Hence, breaking problems down and applying systematic approaches not only helps you find solutions but also reinforces your understanding of underlying physics principles.