HC VERMA PART 1 PAGE 77 OBJECTIVE-1 NEWTON’S LAW OF MOTION-> QUESTION 5 AND QUESTION 6

THE PROBLEM HERE IS THAT WHEN I RESOLVE THE COMPONENTS BY TAKING mg as Y-AXIS THEN the normal component comes out to be mg/cos*(theta). If you resolve the components by taking normal as the y-axis then the normal comes out to be mgcos*(theta). I understand vertical component would be mg/sin* for (i) and mgsin* for(ii) but if we need to find normal force then it will be confusing. This is very confusinmg for me. Please help me out. When I asked some of the good students in my coaching they also didn’t knew it why as they never thought about this. We learnt through vectors that component would be mgcos* and mgsin*. that’s why many didn’t observed it.. Please help..

It sounds like you're grappling with the nuances of resolving forces in inclined plane problems, which can indeed be tricky. Let's break this down step by step to clarify how to approach these types of problems and how to correctly identify the normal force and other components of weight.

Understanding Forces on an Inclined Plane

When dealing with an object resting on an inclined plane, we need to consider the forces acting on it. The two primary forces are the gravitational force (weight) acting downward and the normal force acting perpendicular to the surface of the incline. The gravitational force can be resolved into two components: one parallel to the incline and one perpendicular to it.

Resolving Weight into Components

Let’s denote the weight of the object as mg, where m is the mass and g is the acceleration due to gravity. When you have an incline at an angle θ, you can resolve the weight into two components:

• The component perpendicular to the incline: mg cos(θ)
• The component parallel to the incline: mg sin(θ)

These components are derived from basic trigonometry. The angle θ is the angle between the incline and the horizontal. When you visualize the triangle formed by these components, the adjacent side (perpendicular to the incline) corresponds to mg cos(θ), and the opposite side (parallel to the incline) corresponds to mg sin(θ).

Normal Force Calculation

The normal force is the force exerted by the surface of the incline that supports the weight of the object and acts perpendicular to the surface. In a static situation (where the object is not moving), the normal force N balances the perpendicular component of the weight. Therefore, we can express this relationship as:

N = mg cos(θ)

This means that if you take the weight's perpendicular component as mg cos(θ), the normal force will equal this component when the object is at rest on the incline.

Choosing the Axis for Resolution

Now, regarding your confusion about resolving components using different axes: it’s essential to maintain consistency in your approach. If you choose to take the incline as your reference (where the incline itself is the x-axis), then the normal force will always be mg cos(θ). If you were to take the vertical direction as your y-axis, you would need to account for the incline's angle differently, which can lead to confusion.

In summary, always resolve the weight into components relative to the incline. This way, you can consistently find the normal force as mg cos(θ) and the parallel component as mg sin(θ). If you stick to this method, it will help you avoid the confusion that arises from switching axes.

Practical Example

Let’s consider a practical example. Imagine a block of mass 10 kg resting on a frictionless incline of 30 degrees. The weight of the block is:

W = mg = 10 kg × 9.8 m/s² = 98 N

Now, resolving this weight:

• Perpendicular component: N = mg cos(30°) = 98 N × (√3/2) ≈ 84.87 N
• Parallel component: F_parallel = mg sin(30°) = 98 N × (1/2) = 49 N

Here, the normal force is approximately 84.87 N, which balances the perpendicular component of the weight, while the parallel component would cause the block to slide down the incline if friction were absent.

By consistently applying this method, you can confidently analyze similar problems without confusion. If you have any further questions or need clarification on specific points, feel free to ask!