Question:

A mass of 4 Kg rests on a horizontal plane.The plane is gradually inclined until an angle 15^o with the horizontal, the mass just begins to slide.What is the coefficient of static friction between the block and the surface.

Solution:

Mass of the block 4 Kg

Angle of inclination $${\theta=15^\circ}$$

At a particular angle of inclination the body slide down.

$${F_L \propto N}$$

$${F_L=\mu_s N}$$

$${F_L=\mu_s mg\cos\theta}$$

$${mg\sin\theta=\mu_s mg\cos\theta}$$

$${\frac{\sin\theta}{\cos\theta}=\mu_s}$$

$${\tan 15^\circ=\mu_s}$$

$${\mu_s=0.25}$$

Inclined plane:

A block is placed on an inclined plane and gradually increases the angle of the plane until the  block starts to slide.The force required to overcome static friction is equal to the weight  of the block multiplied by the  sine of angle at which the block starts to move.

Static friction- Key characteristics

• Static friction is directed. (acts parallel to the surfaces in contact, always acting in the opposite direction of the applied force and opposing the direction of imminent motion).
• The magnitude of static friction changes to match the applied force up to a particular limit. Maximum static friction is obtained right before the item begins to move.
• When the applied force to move the items surpasses the maximum static force friction, the frictional force transforms into kinetic friction.
• Static friction friction is determined by the type of the surface in contact as well as the force applied perpendicular to the surfaces in contact (Normal force), and is independent of the area of contact.

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.