Problem 73 - Master Resource Book in JEE Main Physics

Question

A force is inclined at \(60^{\circ}\) to the horizontal. If its rectangular component in the horizontal direction is \(50 \mathrm{~N}\), then magnitude of the force in the vertical direction is (a) \(25 \mathrm{~N}\) (b) \(75 \mathrm{~N}\) (c) \(87 \mathrm{~N}\) (d) \(100 \mathrm{~N}\)

Step-by-Step Solution

Answer

(c) 87 N

Step 1: Identify Given Information

We are told that the force is inclined at an angle of \(60^{\circ}\) to the horizontal, and its horizontal component is \(50 \mathrm{~N}\). We need to find the vertical component of the force.

Step 2: Apply the Relationship for Horizontal Force Component

The horizontal component of a force \( F \) inclined at an angle \( \theta \) is given by \( F_x = F \cdot \cos(\theta) \).Here \( F_x = 50 \mathrm{~N} \) and \( \theta = 60^{\circ} \).

Step 3: Express Force in Terms of its Components

The equation \( F \cdot \cos(60^{\circ}) = 50 \mathrm{~N} \) allows us to solve for \( F \):\[ F = \frac{50}{\cos(60^{\circ})} \]

Step 4: Calculate Force using Trigonometric Function

We know \( \cos(60^{\circ}) = 0.5 \). Substitute this value in:\[ F = \frac{50}{0.5} = 100 \mathrm{~N} \]

Step 5: Find the Vertical Force Component

The vertical component of the force is given by \( F_y = F \cdot \sin(60^{\circ}) \).Substitute \( F = 100 \mathrm{~N} \) and \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \) into the equation:\[ F_y = 100 \cdot \frac{\sqrt{3}}{2} \approx 86.6 \mathrm{~N} \]

Step 6: Select Closest Option

The calculated vertical force component is approximately \(86.6 \mathrm{~N}\), closest to option (c) \(87 \mathrm{~N}\).

Key Concepts

Trigonometry in Physics

In physics, trigonometry helps us understand the relationships between different components of a force. Forces are vectors, and they can be resolved into horizontal and vertical components if they act at an angle. Knowing the angle of a force with respect to a reference line like the horizontal, allows one to use trigonometric functions such as sine and cosine to find these components.

For instance, if a force makes an angle \( \theta \) with the horizontal, the horizontal component of the force \( F \), can be found using the cosine function, as \( F_x = F \cdot \cos(\theta) \). Similarly, the vertical component can be determined using the sine function as \( F_y = F \cdot \sin(\theta) \).

• Cosine is used to find the horizontal component.
• Sine is used to find the vertical component.

Understanding these basic trigonometric relationships is crucial in physics as they enable us to break down complex force interactions into simpler, more manageable parts.

Vector Decomposition

Vector decomposition is a fundamental technique in physics that enables the breaking down of a force acting at an angle into its perpendicular components. This technique simplifies the analysis of forces as they act in different directions. When dealing with forces, it is often more convenient to consider their horizontal and vertical components separately rather than the full inclined force.

The original problem involves a force inclined at \( 60^{\circ} \) with the horizontal. To find both the horizontal and vertical components, one needs to decompose the vector. As per the step-by-step solution, the horizontal component is already given as \( 50 \mathrm{~N} \). Using the trigonometric relationship: \[ F = \frac{F_x}{\cos(60^{\circ})} \] this determines the total magnitude of the force. Then, using the sine function: \[ F_y = F \cdot \sin(60^{\circ}) \] the vertical component is derived.

By following these steps, students can effectively interpret and solve problems involving forces that do not align perfectly along one axis.

Inclined Forces

Forces acting on an incline compose a significant topic in physics, emphasizing the significance of understanding the influence of angles on force components. When a force is inclined, it doesn't act purely in one direction, but shares its effect across perpendicular axes (typically horizontal and vertical).

To analyze such forces, one must consider the effects along the incline's axes. In the given problem, the force angle of \( 60^{\circ} \) resulted in a known horizontal component. This is often the approach in analyzing inclined forces - determine the known component and use trigonometry to resolve the unknown.

• Inclined plane problems are simplified through understanding trigonometric applications.
• Knowledge of angle and one force component leads to determining the other component.

Breaking down an inclined force into components is essential for problems like blocks on ramps or cars on a hill, making this concept both practical and pivotal in real-world physics scenarios.