Problem 16 - Master Resource Book in JEE Main Physics

Question

A car of mass \(1000 \mathrm{~kg}\) moves at a constant speed of \(20 \mathrm{~ms}^{-1}\) up an incline. Assume that the frictional force is \(200 \mathrm{~N}\) and that \(\sin \theta=1 / 20\), where, \(\theta\) is the angle of the incline to the horizontal. The \(g=10 \mathrm{~ms}^{-2}\). Find the power developed by the engine? (a) \(14 \mathrm{~kW}\) (b) \(4 \mathrm{~kW}\) (cl \(10 \mathrm{~kW}\) (d) \(28 \mathrm{~kW}\)

Step-by-Step Solution

Answer

The power developed by the engine is 14 kW (option a).

Step 1: Determine the Gravitational Force Component

Calculate the gravitational force component acting along the incline. The gravitational force component can be calculated using the formula: \[ F_{\text{gravity}} = mg \sin \theta \] where \( m = 1000 \, \text{kg} \), \( g = 10 \, \text{m/s}^2 \), and \( \sin \theta = \frac{1}{20} \). Substitute these values into the formula: \[ F_{\text{gravity}} = 1000 \times 10 \times \frac{1}{20} = 500 \, \text{N} \].

Step 2: Calculate the Total Force Against the Incline

To find the total force against which the engine must work, sum the components of friction and gravitational force. The formula to use is: \[ F_{\text{total}} = F_{\text{friction}} + F_{\text{gravity}} \]. We know \( F_{\text{friction}} = 200 \, \text{N} \) and \( F_{\text{gravity}} = 500 \, \text{N} \). Substitute these values in: \[ F_{\text{total}} = 200 + 500 = 700 \, \text{N} \].

Step 3: Calculate the Power Developed by the Engine

The power developed by the engine can be calculated using the formula: \[ P = F_{\text{total}} \times v \] where \( F_{\text{total}} = 700 \, \text{N} \) and \( v = 20 \, \text{m/s} \). Substitute these values into the formula: \[ P = 700 \times 20 = 14,000 \, \text{W} \]. Convert this value to kilowatts (kW) by dividing by 1000: \[ P = 14,000 \, \text{W} = 14 \, \text{kW} \].

Key Concepts

Inclined Plane Physics

Inclined planes are a fundamental concept in physics, often used to simplify the analysis of objects moving on a slope. An inclined plane is essentially a flat surface tilted at an angle to the horizontal. This angle, typically denoted by \( \theta \), alters the way forces, such as gravity, act on objects moving along the plane.

The analysis of movement on an inclined plane requires breaking down the forces involved.

• Inclined plane physics involves calculating the effective force along the incline and the normal force perpendicular to it.
• The force of gravity acts consistently downward, but its component along the inclined direction is crucial for understanding motion on such planes.

By resolving these gravitational forces, you understand how an object accelerates or remains at a constant speed on an incline. The angle of the incline, depicted by \( \theta \), determines how much of the gravitational force contributes to the movement along the plane. This fundamental analysis is key to solving problems involving inclined planes.

Friction Force Calculation

Friction plays a crucial role in motion on inclined planes. It acts to resist the movement of an object, making it an essential factor in force calculations. In an inclined plane scenario like our exercise, the frictional force must be considered chiefly because it influences the total force or power required to maintain motion.

When calculating friction, you must understand that:

• Frictional force, denoted as \( F_{\text{friction}} \), is typically constant and acts opposite to the direction of intended movement.
• In many problems, this force is given directly or can be computed using the coefficient of friction if further information is provided.

In the provided scenario, knowing the frictional force allows for calculating the total resistance against the motion. This forms the sum with the gravitational force component to give the total force needed by the engine to maintain speed. Recognizing friction's role is pivotal in understanding real-world applications, such as car dynamics on inclines.

Gravitational Force Component

The gravitational force component along an incline is a simplified yet integral part of analyzing motion on a slope. The gravitational force acting on an object can be broken into two components: parallel and perpendicular to the plane. The parallel component, which concerns us in inclined plane physics, is the one responsible for sliding down the incline.

To compute this force:

• Use the formula \( F_{\text{gravity}} = mg \sin \theta \), where \( m \) is mass and \( g \) is the gravitational acceleration.
• The sine of the angle \( \theta \) indicates how much of the weight contributes directly to downward movement along the plane.

This force component is crucial because it dictates how much force the engine or applied effort must counteract to maintain or alter the motion. With \( \theta \) expressed as a ratio like \( \sin \theta = \frac{1}{20} \), as given, it further aids in these straightforward calculations. Understanding these components clarifies how objects behave under gravity on any kind of slope or ramp.