Problem 2 - Master Resource Book in JEE Main Physics

Question

The work done in pulling up a block of wood weighing \(2 \mathrm{kN}\) for a length of \(10 \mathrm{~m}\) on a smooth plane inclined at an angle of \(15^{\circ}\) with the horizontal is \(\left[\sin 15^{\circ}=0.2588\right]\) (a) \(4.36 \mathrm{~kJ}\) (b) \(5.13 \mathrm{k}]\) (c) \(8.91 \mathrm{~kJ}\) (d) \(9.82 \mathrm{~kJ}\)

Step-by-Step Solution

Answer

The work done in pulling up the block is approximately 5.13 kJ (option b).

Step 1: Identify the Forces

In this problem, the work done is calculated by determining the component of the force acting along the inclined plane. The force here is due to gravity acting on the block, which has to be pulled up the incline. The weight of the block is given as \(2 \mathrm{kN}\).

Step 2: Find the Component of Force Along the Incline

To find the component of the weight along the inclined plane, multiply the total weight by the sine of the angle of inclination. Thus, the force component along the plane is \(2 \times \sin 15^\circ = 2 \times 0.2588 = 0.5176 \mathrm{kN}\).

Step 3: Calculate the Work Done

Work done \(W\) is calculated as the product of the force component along the incline and the distance moved along the incline. Here, \(W = 0.5176 \times 10 = 5.176 \mathrm{kJ}\).

Step 4: Compare with Given Options

Match the result with the options provided. The calculated work done is \(5.176 \mathrm{kJ}\), which is closest to option \(b\), \(5.13 \mathrm{kJ}\). Therefore, the correct choice is option \(b\).

Key Concepts

Force Components

When analyzing problems involving inclined planes, it's essential to break down forces into components. This concept helps in understanding the individual effects each force has in a given direction. Here are some key points:

• Forces acting on an object on an inclined plane usually include gravity, normal force, and any applied forces.
• The force of gravity can be split into two components: parallel and perpendicular to the plane.
• The perpendicular component, often counteracted by the normal force, holds no direct effect on movement along the inclined plane.
• The parallel component is calculated as the product of the weight and the sine of the angle of inclination. This force component causes the object to slide down, or makes it necessary to apply a force to move the object up the incline.

By focusing on these components, calculations become more straightforward, allowing for accurate determination of work or energy changes in systems involving inclined planes.

Inclined Plane Physics

An inclined plane is a flat surface tilted at an angle to the horizontal, and it is a classic problem often used to illustrate fundamental concepts in physics. In scenarios involving inclined planes, several key concepts are crucial:

• The angle of inclination plays a critical role in determining the effective forces acting on an object on the plane.
• Motion on an inclined plane involves the interplay of force components, where understanding their directional influence is key.
• Since the plane is smooth in many academic exercises, it is presumed to have no friction, simplifying calculations.
• The work done by forces on an inclined plane can be calculated using the formula \(W = F \cdot d\), where \(F\) is the force component along the plane, and \(d\) is the distance moved along the plane.

Understanding these principles helps to predict and calculate the motion dynamics and energy changes in problems involving inclined planes.

Trigonometry in Physics

Trigonometry is often the backbone in solving physics problems, especially those involving inclined planes or angles. Here's why trigonometry is vital:

• It allows the decomposition of forces into components, which is crucial in multi-dimensional motion analysis.
• Key trigonometric functions, such as sine, cosine, and tangent, are used to find relationships between angles and lengths in such setups.
• For force components, the angle of inclination is used with sine and cosine to find the parallel and perpendicular forces, respectively.
• In the problem under discussion, the calculation of the force component along the incline uses \(F_{\parallel} = F \cdot \sin(\theta)\), which simplifies work and energy calculations on the plane.

These trigonometric relationships make it possible to analyze complex systems with simple mathematical expressions, making them indispensable in physics.