Problem 350 - JEE Physics

Question

A given object takes n times more time to slide down \(45^{\circ}\) rough inclined plane as it takes to slide down a perfectly smooth \(45^{\circ}\) incline. The coefficient of kinetic friction between the object and the incline is (A) \(\left[1 /\left(2-\mathrm{n}^{2}\right)\right]\) (B) \(\left[1-\left(1 / \mathrm{n}^{2}\right)\right]\)

Step-by-Step Solution

Answer

The coefficient of kinetic friction between the object and the incline is (B) \(\left[1-\left(1 / \mathrm{n}^{2}\right)\right]\).

Step 1: Analyze Forces Acting on Object

First, let's analyze the forces acting on the object in both scenarios. We have two main forces acting on the object: gravity and friction (in the case of a rough incline). The force of gravity has two components: one is parallel to the incline, and the other is perpendicular to the incline. The force of friction is opposite to the motion of the object and is directly proportional to the force pressing the object and the plane together (normal force).

Step 2: Equations for Smooth Inclined Plane

In the case of a smooth inclined plane, there is no friction force acting on the object. The only force causing the object to slide down the incline is the gravitational force parallel to the incline. This parallel force can be found using the equation: \(F_{g \parallel} = mg \sin \theta\), where m is the mass of the object, g is the gravitational acceleration, and \(\theta\) is the angle of inclination. Since the angle of inclination is 45 degrees, this equation simplifies to: \(F_{g \parallel} = mg \sin 45^{\circ} = mg/\sqrt{2}\) Using Newton's Second Law of Motion, the acceleration of the object on the smooth incline can be found: \(F_{g \parallel} = m a_{smooth}\), which simplifies to: \(a_{smooth} = g/\sqrt{2}\)

Step 3: Equations for Rough Inclined Plane

In the case of a rough inclined plane, there is friction force acting on the object. The friction force can be found using the equation: \(F_{friction} = \mu F_{normal}\), where \(\mu\) is the coefficient of kinetic friction and \(F_{normal}\) is the normal force between the object and the incline. The force acting perpendicular to the plane is equal to the normal force in this case: \(F_{normal} = mg \cos \theta\), which simplifies to: \(F_{normal} = mg/\sqrt{2}\) Combining both equations and substituting back the force due to gravity component, \(F_{friction} = \mu (mg/\sqrt{2})\) Now, the net force acting on the object along the incline is the difference between the gravitational force parallel to the incline and the friction force. Applying Newton's Second Law of Motion: \(F_{g \parallel} - F_{friction} = m a_{rough}\) Substituting both forces, we get: \((mg/\sqrt{2}) - (\mu (mg/\sqrt{2})) = ma_{rough}\) This simplifies to: \((1-\mu)(g/\sqrt{2}) = a_{rough}\)

Step 4: Finding Time Ratio

According to the problem, the object takes n times more time to slide down the rough incline than it takes to slide down the smooth incline. Therefore, the time taken in both cases can be represented as: \(t_{rough} = n t_{smooth}\) The distance traveled down the incline in both cases is the same. Using the equation of motion, we have: \(d = \frac{1}{2} a t^2\) Since distance d is the same, we can equate the smooth and rough situations: \(\frac{1}{2} a_{smooth} t_{smooth}^2 = \frac{1}{2} a_{rough} t_{rough}^2\)

Step 5: Solve for Coefficient of Kinetic Friction

Now, substitute the expressions of time and accelerations from Step 3 and Step 4 into the equation above: \(\frac{1}{2} (g/\sqrt{2}) t_{smooth}^2 = \frac{1}{2} ((1-\mu)(g/\sqrt{2})) (n t_{smooth})^2\) Simplifying and canceling terms, we can solve for the coefficient of kinetic friction: \(\mu = 1 - \frac{1}{n^2}\) Therefore, the answer is (B) \(\left[1-\left(1 / \mathrm{n}^{2}\right)\right]\).

Key Concepts

Newton's Second Law

The core principle governing the movement of objects is Newton's Second Law of Motion. It tells us that the acceleration of an object depends on the net force acting upon it and the mass of the object. In simpler terms, it's given by the formula: \[ F = ma \]where:

• \( F \) is the net force applied,
• \( m \) is the mass of the object, and
• \( a \) is the acceleration.

On an inclined plane, when an object slides down, this law helps us calculate how quickly it will accelerate based on the forces acting on it. For instance, on a smooth inclined plane, only gravity acts on the object in the direction of motion, allowing us to compute acceleration easily using Newton's Second Law. The object's acceleration can be expressed as:\[ a = \frac{F_{net}}{m} \] where the net force is the gravitational component parallel to the incline.

Coefficient of Kinetic Friction

When dealing with an inclined plane that is rough, friction plays a crucial role. The coefficient of kinetic friction, denoted by \( \mu \), quantifies how much frictional force acts on the sliding object. It determines how resistant the surface is to motion, thereby affecting the object's acceleration.
The frictional force can be calculated using:\[ F_{friction} = \mu F_{normal} \] where:

• \( F_{friction} \) is the force of friction,
• \( \mu \) is the coefficient of kinetic friction, and
• \( F_{normal} \) is the normal force, or the force perpendicular to the plane.

Understanding the coefficient of kinetic friction helps us predict how much more difficult it is for an object to slide down an inclined plane compared to a smooth one.

Gravitational Force Components

The influence of gravity on an inclined plane can be decomposed into two main components: gravitational force parallel to the slope and gravitational force perpendicular to it.

• The parallel component, \( F_{g \parallel} \), pulls the object down the incline. It's calculated as: \[ F_{g \parallel} = mg \sin \theta \]where \( \theta \) is the angle of slope.
• The perpendicular component, \( F_{g \perp} \), presses the object onto the plane. It's given by:\[ F_{g \perp} = mg \cos \theta \]This component is crucial as it helps us determine the normal force.

Together, these forces help determine the net force acting on the object, affecting its acceleration.

Equations of Motion

In physics, equations of motion describe the movement of objects. For an inclined plane, these equations help calculate how long and how swiftly an object will move.
One of the key equations we use is:\[ d = \frac{1}{2} at^2 \] where:

• \( d \) is the distance traveled,
• \( a \) is the acceleration, and
• \( t \) is the time elapsed.

In our scenario, where an object descends both a smooth and a rough incline, this formula proves vital in comparing the time taken to travel the same distance with differing frictional effects. By equating the distances from both scenarios, we can deduce relationships between timing, friction, and acceleration.