An object is pushed up along a plane of inclination 30 degree.It comes down to the initial point after sometime with half the initial upward velocity.Find coefficient of sliding friction between the body and plane.

To determine the coefficient of sliding friction between the object and the inclined plane, we can analyze the forces acting on the object as it moves up and then slides back down. Let's break this down step by step.

Understanding the Forces at Play

When the object is pushed up the incline, several forces are acting on it:

• Gravitational Force (Weight): This acts downward and can be resolved into two components: one perpendicular to the incline and one parallel to it.
• Normal Force: This acts perpendicular to the surface of the incline and balances the perpendicular component of the weight.
• Frictional Force: This opposes the motion of the object and acts down the incline when the object is moving up.

Setting Up the Problem

Let’s denote the following:

• m: mass of the object
• g: acceleration due to gravity (approximately 9.81 m/s²)
• θ: angle of inclination (30 degrees)
• u: initial upward velocity
• v: final downward velocity (which is half of the initial velocity, so v = u/2)

Analyzing the Motion Up the Incline

When the object is pushed up, it experiences a net force that can be expressed as:

Net Force = Applied Force - Friction - Gravitational Component along the incline

The gravitational component acting down the incline is given by:

F_gravity = mg sin(θ)

The normal force (N) is:

N = mg cos(θ)

The frictional force (F_friction) can be expressed as:

F_friction = μN = μmg cos(θ)

Where μ is the coefficient of friction.

Using Kinematics to Relate Forces

As the object moves up the incline, it eventually comes to a stop and then slides back down. The distance it travels up can be calculated using the kinematic equation:

v² = u² + 2a s

Where:

• v = final velocity (0 at the top)
• u = initial velocity
• a = acceleration (which will be negative due to gravity and friction)
• s = distance traveled up the incline

Rearranging gives us:

0 = u² + 2(-g sin(θ) - μg cos(θ)) s

Considering the Descent

When the object slides back down, it starts with a velocity of 0 and accelerates down the incline. The net force acting on it while descending is:

Net Force = Gravitational Component - Friction

Using the same kinematic equation, we can express the motion down the incline:

v² = 0 + 2(g sin(θ) - μg cos(θ)) s

Substituting v = u/2 gives:

(u/2)² = 2(g sin(θ) - μg cos(θ)) s

Equating the Two Scenarios

Since the distance traveled up and down is the same, we can set the two equations equal to each other. By solving these equations simultaneously, we can isolate μ:

From the upward motion, we can express s in terms of u, g, θ, and μ. Then, substituting this expression into the downward motion equation allows us to solve for μ.

Final Calculation

After some algebra, we find that:

μ = (u² / (4g cos(θ))) - (1/2) tan(θ)

Substituting θ = 30 degrees (where sin(30) = 0.5 and cos(30) = √3/2) into the equation gives us:

μ = (u² / (4 * 9.81 * (√3/2))) - (1/2 * (1/√3))

Calculating this will yield the coefficient of friction between the object and the inclined plane.

In conclusion, by analyzing the forces and applying kinematic equations, we can derive the coefficient of sliding friction effectively. This approach not only helps in solving the problem but also deepens our understanding of the dynamics involved in inclined plane motion.