Question

Which has a greater horizontal component, a 100-N force directed at an angle of \(60^{\circ}\) above the horizontal or a 60-N force directed at an angle of \(30^{\circ}\) above the horizontal?

Short answer

Answer: The 60-N force at a 30-degree angle above the horizontal has a greater horizontal component (approximately 51.96 N) compared to the 100-N force at a 60-degree angle above the horizontal (approximately 50 N).

Step-by-step solution

Determine the Formula and Variables

We will use the following formula to find the horizontal component (x-component) of a force (F), given the angle (θ) with respect to the horizontal: Horizontal component (H) = F × cos(θ) We have two force vectors: - A 100-N force at a 60-degree angle above the horizontal. - A 60-N force at a 30-degree angle above the horizontal. Our task is to find the horizontal components of these forces and compare their magnitudes.

Calculate the Horizontal Component of the 100-N Force

For the 100-N force, we have: F1 = 100 N θ1 = 60 degrees First, we'll need to convert the angle from degrees to radians: θ1 = (60 * π) / 180 = (π / 3) radians Now we can use the formula to calculate the first horizontal component, H1: H1 = F1 × cos(θ1) H1 = 100 × cos(π / 3)

Calculate the Horizontal Component of the 60-N Force

For the 60-N force, we have: F2 = 60 N θ2 = 30 degrees Again, we'll need to convert the angle from degrees to radians: θ2 = (30 * π) / 180 = (π / 6) radians Now we can use the formula to calculate the second horizontal component, H2: H2 = F2 × cos(θ2) H2 = 60 × cos(π / 6)

Compare the Magnitudes of the Horizontal Components

Now we compare the magnitudes of the horizontal components, H1 and H2: H1 = 100 × cos(π / 3) = 50 N (approximate) H2 = 60 × cos(π / 6) = 51.96 N (approximate) Therefore, the horizontal component of the 60-N force directed at a 30-degree angle above the horizontal (approximately 51.96 N) is greater than the horizontal component of the 100-N force directed at a 60-degree angle above the horizontal (approximately 50 N).

Key concepts

Force Vector Analysis

Force vector analysis is a fundamental part of physics, particularly in the field of mechanics. It is the process of breaking down a force into its components, which typically include a horizontal (x-axis) and a vertical (y-axis) component. This breakdown is crucial as it allows physicists and engineers to analyze forces more effectively in various applications, such as determining the stress on a structure or the acceleration of an object.

When dealing with force vectors, it is often necessary to calculate the magnitude of the force in a specific direction. This is where trigonometry comes into play. By using trigonometric functions, such as the sine and cosine, we can resolve a force into its components based on the angle it forms with a given axis. In the case of the horizontal component of the force, we would use the cosine function.

Remember, the horizontal component of any force vector can be found using the formula:
\( H = F \times \cos(\theta) \), where \( H \) is the horizontal component, \( F \) is the magnitude of the force, and \( \theta \) is the angle the force makes with the horizontal.

Trigonometry in Physics

Trigonometry is not just a branch of mathematics; it is a powerful tool used extensively in physics to solve problems involving angles and distances. This branch of mathematics deals with the relationships between the sides and angles of triangles. In physics, we commonly use it to understand the relationships between different components of vectors, such as displacement, velocity, force, and more.

In the case of force vectors, trigonometric functions help us determine the components along the x- and y-axes. For example, the cosine function is used to calculate the horizontal component when the force's angle with the horizontal line is known.

The formula involving the cosine function to find the horizontal component is highly relevant, as it allows us to project the force vector onto the x-axis. In practical terms, this enables us to understand how much of a force is 'pushing' in the horizontal direction—a critical piece of information when solving for equilibrium or motion in physics.

Converting Degrees to Radians

In physics, as in many areas of mathematics, it's important to be able to convert between degrees and radians. Degrees are often used in everyday language and when measuring angles in a more general context. However, radians are the standard unit of angular measure used in the field of mathematics because they provide a direct link between the angle and the arc length of a circle.

All trigonometric functions in calculus are typically defined in terms of radians, and this is why converting angles from degrees to radians is essential in trigonometry and physics. The conversion formula is quite simple:
\( \text{{radians}} = \text{{degrees}} \times \frac{\pi}{180} \).

Therefore, to work with trigonometric functions in physics problems, such as finding the horizontal component of a force vector, one must become comfortable with this conversion. It ensures accurate calculations and helps to internalize the relationship between the various units of measurement used to describe rotational motion and angles.