Question:

A mass of 2 Kg lies on an inclined plane as shown below.Resolve its weight along and perpendicular to the plane. (take g=10 ms^-2 )

Solution: Inclined plane

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Resolving the weight into two components

The component which is along the plane is $${mg\sin\theta}$$

$${20 \sin\ 30^\circ}$$

$${20 (\frac{1}{2})}$$

$${=10 \ N}$$

The component which is perpendicular to the plane is $${mg\cos\theta}$$

$${20 \cos\ 30^\circ}$$

$${20 (\frac{\sqrt{3}} {2})}$$

$${10\sqrt{3} \ N}$$

Key concepts in ‘ The mass in an inclined plane’:

Newton’s Laws of Motion:

Newton’s laws govern the relationship between force, mass, and acceleration. The forces acting on the mass are Gravitational force,Normal force and frictional force.

Vector Resolution:

The gravitational force can be split into parallel component (causes the object to slide down) and the perpendicular component (acts perpendicular to the surface and is balanced by the normal force.

Equilibrium and motion:

When the object is at rest,the net force acting on the plane is zero (called the static equilibrium).When the object moves with constant velocity, the net force is still zero (called the dynamic equilibrium).

Acceleration is calculated using the ‘Newton’s second law’ F= ma, when the object is accelerating.

Friction:

Understanding the types of friction (static and kinetic) and their effects on motion.

Inclined Plane Geometry:

Using trigonometry to relate the angle of inclination to the force components.

Energy Considerations:

Potential energy is stored due to the object’s height above the base of the inclined plane and is given by U=mgh

Kinetic energy is due to the object’s motion and is given by K=1/2mv²

Work-energy principle is obeyed where the work done by the forces results in a change in the object’s kinetic and potential energies.

Real Life applications of ‘The mass in an inclined plane’

Inclined planes are fundamental to many real world systems.

Ramps and slopes:

Used for wheelchairs,loading docks and skateparks to overcome height differences with less effort.

Roads and railways:

Roads and railways are often inclinedto manage elevation changes in a gradual manner, ensuring safety and ease of transportation.

Construction and Architecture:

Inclined planes are used in designing stairs, escalators and conveyor systems.

Roofs:

The slope of a roof affects the runoff of rainwater and snow.

Machinery and Tools:

Inclined planes are fundamental in tools like wedges and screw jacks, which help in lifting or splitting objects by converting a small input force into a larger output force.