Question:

A block of mass m slides down the plane inclined at an angle 60^o with an acceleration g/2 .Find the coefficient of kinetic friction.

Solution: Finding the coefficient of friction

As the block of mass m is moving on the inclined plane , the kinetic friction f_k comes into play.

The forces acting on the mass m are

• Gravitational force (mg)
• Normal force (N)

Gravitational force (mg):

We can resolve the gravitational force into two components.The component which is along the plane is $${mg \ \sin\theta}$$

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

Normal force(N):

The normal force is exerted by the inclined plane on the block of mass m.This normal force is perpendicular to the plane and is balanced by the perpendicular component of the force.

$${mg \ \cos\theta=N}$$

Frictional force (f_k):

The frictional force opposes the motion of the block.

$${f_k\propto N}$$

$${f_k=\mu_k N}$$

The coefficient of kinetic friction $${\mu_k}$$ is a dimensionless number, depends on the nature and conditions of two surfaces.The value of the coefficient of kinetic friction ranges from 0.1 to 1.5 and is independent of the speed.

Along the plane:

The equation of motion along the plane, inclined at an angle of 60 degree and the acceleration a=g/2 and the net force acting along the plane is

$${mg \ \sin\theta-f_k =ma}$$

$${mg \ \sin 60^\circ -f_k=ma}$$

$${mg\frac{\sqrt3}{2}-f_k=m(\frac{g}{2})}$$

$${f_k=mg(\frac{\sqrt3-1}{2})}$$

Perpendicular to the plane:

There is no motion along the perpendicular direction as the normal force and the perpendicular component of the force are balanced each other.

$${mg \ \cos\theta =N}$$

substitute $${f_k=\mu_k N}$$ and $${N=mg \ \cos\theta}$$

$${mg \ \sin\theta-\mu_k (mg \ cos\theta) =m(\frac{g}{2})}$$

$${\sin\theta -\mu_k \cos\theta=\frac {1}{2}}$$

$${\sin 60^\circ-\mu_k \cos 60^\circ=\frac {1}{2}}$$

$${\frac{\sqrt3}{2}-\mu_k(\frac{1}{2})=\frac{1}{2}}$$

$${\sqrt3 – \mu_k=1}$$

$${\mu_k=\sqrt3-1=1.732-1=0.732}$$

Friction in an inclined plane:

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’:

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.