Inclined Plane

Inclined Plane is the most fundamental forms of mechanical devices used in physics. In order to get around physical obstacles and simplify tasks, inclined planes have been used for centuries in both ancient and recent construction projects. A flat surface that is angled with respect to the horizontal plane is the fundamental component of an inclined plane. It is a basic mechanism that works by extending the force over a greater distance in order to decrease the force required to move an object vertically.

In this article, we will learn in detail about inclined plane, the mechanics behind it, the resolution of forces into horizontal perpendicular component acting on inclined plane and solve examples based on it.

What is Inclined Plane?

Inclined Plane in physics can be considered as basic machine with a flat surface angled at an angle towards the horizontal plane. It offers an approach of lifting objects while using less force than necessary to raise them vertically. There are many applications of inclined planes such as ramps, hills, and even the inclined surfaces of staircases. They spread the labour over a greater area, enabling things to be moved up or down with less force.

Mechanics of Inclined Planes

An inclined plane’s main function is to promote the shifting of items vertically by applying force horizontally. The longer the inclined plane, the less force is required to lift an object. This principle is fundamental to understanding how inclined planes work and why they are useful in various applications.

When an object needs to be moved vertically, such as lifting a box onto a shelf, the force required to overcome gravity directly is significant. However, by using an inclined plane, the force needed to lift the object vertically can be reduced. Here’s how the mechanics of inclined planes work:

Force Distribution: When an object is placed on an inclined plane, the weight of the object acting downwards results in one component of the force of gravity acting on the object to be parallel to the surface of the inclined plane, and the normal force exerted by the inclined plane upwards is the other component that can be resolved into two parts. It is possible to separate the object’s weight into two parts: one that is parallel to the inclined plane and the other that is perpendicular to it.

Reduction of Force: The movement along the inclined plane is distinct of the part of the object’s weight acting perpendicular to it. As a result, the movement is only affected by the weight component parallel to the inclined plane. This indicates that, in comparison to raising the object vertically, less effective force is required to move it along the inclined plane.

Mechanical Advantage: The mechanical advantage of an inclined plane is calculated by finding the ratio of the length of the inclined plane to its height. A longer inclined plane requires less force to move an object along its length. This is because the longer length allows the force to be applied over a greater distance, reducing the force needed to overcome gravity.

Friction: Movement of the object may be affected by friction between it and the inclined plane. By giving friction and stopping the thing from sliding too quickly, friction can be helpful. On the other hand, too much friction can make things difficult to move and need more force to overcome.

Angle of Inclination: The efficiency of the inclined plane also gets affected by the tilt angle. Up to a certain point, the force needed to move an item along a plane decreases as the angle of inclination increases. After that, the effort needed can rise because of the incline’s increased steepness and possible frictional effects.

Normal Force in Inclined Planes

In an inclined plane, the normal force is not directed in the usual direction. Until now, normal force has always been seen to be directed upward in the opposite direction of gravity. The reality about normal forces is that they are always directed perpendicular to the surface the object is on, not always upward.

Components of Normal Force

In a normal plane, which is a horizontal surface, the object’s weight is completely counteracted by the normal force.

N = mg

The normal force is perpendicular to the surface on an inclined plane, and it compensates the perpendicular component of the gravitational force.

if θ represents the angle of inclination, the normal force N can be expressed as:

N = mg cos(θ)

Example:

Imagine a book resting on a ramp. Gravity pulls the book straight down, but because the ramp is slanted, that pull isn’t straight up against our hands. Here, gravity has got two faces. Gravity acts on the book with one big force, but on a slant, that force gets divided into two parts:

• Normal Force: This force pushes the book upward, perpendicular to the ramp’s surface. Its like the ramp pushing back upward.
• Weight Pulling Down the Ramp: This force acts downward along the slope of the ramp it is the unbalanced force that pulls the book down

Gravity Force Components

There are two forces acting on the object, one object is broken down into a perpendicular components so that it can easily added to other forces acting on the object. The force acting on the object is broken down into two components – Horizontal and Vertical.

In inclined planes Concept, we resolve the Force vector (F_grav) into two parts, one directed parallel to the inclined surface and the other directed perpendicular to the inclined surface.

On an inclined plane, the gravitational force can also be resolved into two components:

F_parallel = mgsin(θ){parallel to the incline}

F_{perpendicular} = mgcos(θ){perpendicular to the incline}

Object on an Inclined Plane

If an object of mass m is placed on a smooth inclined plane (i.e. frictional force F = 0) and released it will slide down the slope. To find the acceleration of the particle as it slides we resolve in the direction of motion.

F = ma

mg cos(90 — θ) = ma

g cos(90 — θ) = a

g sin(θ) = a

We can see that the particle’s mass does not affect the acceleration but only the angle of the slope does.

If a particle of mass m is placed on a rough inclined plane (i.e. the frictional force F is not 0), if sliding of F is large enough.

We resolve perpendicular to the plane, where acceleration is zero.

F = ma,

R – mg cos θ = m×0

R = mg cos θ

We resolve in the direction of the slope, if the particle is at rest then

F = ma

mg cos(90 – θ) – F = m × 0

mg sin θ = F

Where F is the force of friction. We know that the maximum frictional force is given by F_max = uR

Therefore

F ≤ uR

mg sin(θ) ≤ u mg cos(θ),

sin(θ)/cos(θ)=tan(θ)

tan(θ) ≤ u

Therefore the particle will remain at rest until tan(θ) > u, at this point it will accelerate down the slope.

Solved Examples on Inclined Plane

Example 1: A block of mass 5 kg rests on an inclined plane inclined at an angle of 30 degrees to the horizontal. If the coefficient of friction between the block and the inclined plane is 0.2, calculate:

a) The normal force acting on the block.

b) The frictional force acting on the block if it is on the verge of sliding down.

c) The acceleration of the block if it is released from rest.

Solution:

a) The normal force (N) can be calculated using the formula:

N = mg cos(θ)

where m 5 kg, g =9.81 m/s (acceleration due to gravity), and θ = 30

N (5 kg)(9.81 m/s²) cos(30°)

N = (5)(9.81)(√3/2)

N = 42.73 N

b) The frictional force (f) can be calculated using the formula:

f = uN;

where u= 0.2 (coefficient of friction).

f = (0.2) (42.73)

f = 8.55 N

c) The net force acting on the block (Fnet) when it is released from rest is the component of the

gravitational force parallel to the inclined plane minus the frictional force:

F_net = mg sin(θ) — f

F_net = (5) (9.81) sin(30°) – 8.55

F_net = 24.52 – 8.55

F_net = 15.97 N

The acceleration (a) of the block can be calculated using Newton’s second law (F_net = ma)

15.97 = (5)a

a = 15.97/5

a = 3.194 m/s

Example 2: A 10 kg box is placed on an inclined plane inclined at an angle of 45 degrees to the horizontal. If the coefficient of friction between the box and the inclined plane is 0.3, determine the force parallel to the incline required to move the box up the incline at constant velocity.

Solution:

The force parallel to the incline required to move the box up the incline at constant velocity is

equal to the force of friction acting down the incline.

f=uN

Where u = 0.3 (coefficient of friction).

N = mg cos(θ)

N = (I0 kg)(9.81 m/s²) cos(45°)

N = 69.3N

f = 20.79 N

Example 3: A block of mass 12 kg is placed on an inclined plane inclined at an angle of 30 degrees to the horizontal. If the coefficient of friction between the block and the inclined plane is 0.25, calculate the acceleration of the block when it is released from rest.

Solution:

The net force acting on the block( net) when it is released from rest is the component of the

gravitational force parallel to the inclined plane

F_net = mg sin(θ) — f

f = uN

N = mg cos(θ)

F_net = mg sin(θ) — umg cos(θ)

F_net =m(g sin(0) — ug cos(θ))

F_net = 12(9.81 × sin(30°) – 0.25 × 9.81 × cos(30°))

F_net = 12(4.905 – 0.25 × 8.484)

F_net =12(4.905 – 2.121)

F_net = 12 × 2.784

F_net = 33.408 N

The acceleration (a) of the block can be calculated using Newton’s second law (Fnet=m*a)

33.408 = 12 × a

a = 33.408/12

a = 2.784m s

FAQs on Inclined Plane

What is an inclined plane?

An inclined plane is a simple machine consisting of a flat surface that is tilted at an angle to the horizontal. It allows objects to be moved up or down with less force than lifting them vertically.

How does an inclined plane reduce the force needed to move objects?

An inclined plane reduces the force needed to move objects by spreading the effort required over a longer distance along the inclined surface. This is achieved by converting some of the force needed to lift the object vertically into a force parallel to the plane.

What factors affect the effectiveness of inclined planes?

The effectiveness of inclined planes depends on factors such as the angle of inclination, the coefficient of friction between the object and the inclined surface, and the length of the inclined plane.

What are some common examples of inclined planes in everyday life?

Common examples of inclined planes include ramps, roads, wheelchair ramps, loading ramps for trucks, and sloping driveways.

How is the normal force related to inclined planes?

The normal force is the perpendicular force exerted by the inclined plane on an object resting on it. It is crucial for balancing the component of the object’s weight perpendicular to the plane and ensuring equilibrium.

Can inclined planes be used to move objects horizontally?

Yes, inclined planes can also be used to move objects horizontally. In this case, the force applied parallel to the inclined surface causes the object to slide along the plane, overcoming friction.