Question

The upper half of an inclined plane of inclination \(\theta\) is perfectly smooth while the lower half is rough A body starting from the rest at top come back to rest at the bottom, then the coefficient of friction for the lower half is given by (A) \(\mu=\sin \theta\) (B) \(\mu=\cot \theta\) (C) \(\mu=2 \cos \theta\) (D) \(\mu=2 \tan \theta\)

Short answer

The coefficient of friction (\(\mu\)) for the lower half of the inclined plane is equal to 1. However, none of the given options match this result, so the correct answer is not listed among the given options.

Step-by-step solution

Analyze the motion on the upper (smooth) half of the inclined plane

As the body slides down the smooth half of the inclined plane, it only experiences acceleration due to gravity. The motion is governed by the following equation: \(v^2 = u^2 + 2as\) where, \(v\) is the final velocity reached at the bottom of the smooth half, \(u\) is the initial velocity (0, since the body starts from rest), \(a\) is the acceleration due to gravity along the inclined plane (\(g\sin\theta\)), \(s\) is the distance traveled (half the inclined plane length, let's denote it by \(L/2\)). Substituting these values, we get: \(v^2 = 0^2 + 2(g\sin\theta)(L/2)\) Simplifying the equation: \(v^2 = gL\sin\theta\)

Analyze the motion on the lower (rough) half of the inclined plane

On the lower half of the inclined plane, the body experiences both gravitational force and frictional force. We can write the equation of motion for this segment: \(v'^2 = v^2 - 2\mu a's'\) where, \(v'\) is the final velocity at the bottom of the rough half (0, since the body comes to rest), \(v\) is the initial velocity reached at the end of the smooth half, \(\mu\) is the coefficient of friction between the body and the rough surface, \(a'\) is the acceleration due to gravity along the inclined plane (\(g\sin\theta\)) with the negative sign because it opposes the body's motion, \(s'\) is the distance traveled (the other half of the inclined plane's length, denoted by \(L/2\)). Substituting these values, we get: \(v'^2 = v^2 - 2\mu (g\sin\theta)(L/2)\) Simplifying the equation: \(v'^2 = v^2 - \mu gL\sin\theta\)

Apply the conservation of mechanical energy principle

Since the final velocity \(v'\) at the end of the rough half is 0 (the body comes to rest), we can equate the mechanical energy during smooth segment and rough segment using conservation of mechanical energy: \(gL\sin\theta = \mu gL\sin\theta\) Divide both sides by \(g L\sin\theta\): \(1 = \mu\) Here, we find that the coefficient of friction (\(\mu\)) for the lower half of the inclined plane is equal to 1. Checking the given options, we find that: Option (A) \(\mu=\sin \theta\) Option (B) \(\mu=\cot \theta\) Option (C) \(\mu=2 \cos \theta\) Option (D) \(\mu=2 \tan \theta\) None of the given options match our result (\(\mu = 1\)). Therefore, the correct answer is not listed among the given options.

Key concepts

Friction

Friction is a force that opposes the motion of objects. When you have a surface that is not smooth, like the lower half of our inclined plane, you will experience friction.

• Friction comes from the interaction between the surfaces in contact. The rougher the surface, the greater the friction.
• The strength of friction depends on two things: the roughness of the surfaces and how hard they are pushed together.
• The coefficient of friction (\(\mu\)) is a number that represents how much friction exists between surfaces.
• In our exercise, the frictional force acts to stop the motion as the object moves down the rough part of the inclined plane. It works against the gravitational pull that tries to slide the object further down the plane.

Without friction, the object would have kept rolling indefinitely since nothing would be there to slow it down.

Gravitational Force

Gravitational force is the attraction between two masses, like the Earth and an object. It is what pulls objects down the inclined plane.

• This force is what causes an object to slide downwards on our inclined plane.
• The component of gravitational force that causes the object to move down an inclined plane is given by \(mg\sin\theta\), where \(\theta\) is the angle of the incline, \(m\) is the object's mass, and \(g\) is the acceleration due to gravity (approximately 9.8 m/s²).
• In terms of energy, gravitational force converts potential energy into kinetic energy as the object slides down the plane.

It's fascinating how gravity doesn't just pull objects down but also affects how objects interact with surfaces, giving rise to friction in this scenario.

Mechanical Energy Conservation

The principle of mechanical energy conservation states that in the absence of non-conservative forces (like friction), the total mechanical energy (potential energy + kinetic energy) remains constant. However, in real-world scenarios, some energy is usually lost to friction.

• In the smooth half of the inclined plane, mechanical energy conservation fully applies, meaning no energy is lost. All potential energy converts to kinetic energy.
• Once the object reaches the rough half, friction comes into play, dissipating some of the mechanical energy as heat.
• The exercise uses the equation of energy conservation to find that frictional force brings the object to rest at the bottom, using the same amount of energy gained. This principle assists in calculating the coefficient of friction because it defines how much of the total energy is used by friction.

Even when friction is present, understanding how mechanical energy is transferred helps explain why objects eventually stop moving.

Inclination Angle

The inclination angle (\(\theta\)) of an inclined plane is crucial for analyzing the physics of motion on a slope.

• This angle determines how much of the gravitational force acts along the plane's surface.
• A larger angle means more force helps move the object downward, increasing its acceleration and speed.
• Conversely, a smaller angle results in a lesser force component along the plane, making it harder for the object to slide.
• The angle is not just a geometric property; it affects how energy and forces transform as objects move.

Our problem shows how the angle influences both friction and gravitational components, leading to the analysis of energy conservation as the angle dictates how much energy is transferred versus conserved.