A body takes 9/7times as much time to slide down a rough inclined plane as it takes to slide an identical but smooth inclined plane .find coefficient of friction if the angle of incline is 30 degree

To solve the problem of finding the coefficient of friction for a body sliding down a rough inclined plane compared to a smooth one, we can break it down step by step. The key here is to understand the forces acting on the body and how they relate to the time taken to slide down the incline.

Understanding the Motion on Inclined Planes

When an object slides down an inclined plane, two main forces act on it: gravitational force and frictional force. The gravitational force can be resolved into two components: one parallel to the incline (which causes the object to slide down) and one perpendicular to the incline (which affects the normal force).

Forces Acting on the Body

Let's denote:

• g = acceleration due to gravity (approximately 9.81 m/s²)
• θ = angle of incline (30 degrees)
• μ = coefficient of friction

The gravitational force acting down the incline is given by:

F_parallel = mg sin(θ)

The normal force acting on the body is:

F_normal = mg cos(θ)

The frictional force opposing the motion is:

F_friction = μ F_normal = μ mg cos(θ)

Net Force and Acceleration

The net force acting on the body sliding down the rough incline can be expressed as:

F_net = F_parallel - F_friction = mg sin(θ) - μ mg cos(θ)

Thus, the net acceleration (a) of the body on the rough incline is:

a = g sin(θ) - μ g cos(θ)

Time of Descent

The time taken to slide down an incline can be derived from the kinematic equations. For an incline of length L, the time (t) taken to slide down is given by:

t = √(2L/a)

For the smooth incline, the acceleration is simply:

a_smooth = g sin(θ)

Thus, the time taken to slide down the smooth incline (t_smooth) is:

t_smooth = √(2L/(g sin(θ)))

For the rough incline, substituting the expression for acceleration:

t_rough = √(2L/(g sin(θ) - μ g cos(θ)))

Setting Up the Equation

According to the problem, the time taken to slide down the rough incline is 9/7 times that of the smooth incline:

t_rough = (9/7) t_smooth

Substituting the expressions for time:

√(2L/(g sin(θ) - μ g cos(θ))) = (9/7) √(2L/(g sin(θ)))

Squaring both sides and simplifying gives:

2L/(g sin(θ) - μ g cos(θ)) = (81/49) * 2L/(g sin(θ))

Canceling 2L from both sides and rearranging leads to:

g sin(θ) = (81/49)(g sin(θ) - μ g cos(θ))

Solving for the Coefficient of Friction

Now, we can simplify this equation:

49 g sin(θ) = 81 g sin(θ) - 81 μ g cos(θ)

Rearranging gives:

81 μ g cos(θ) = 81 g sin(θ) - 49 g sin(θ)

81 μ g cos(θ) = 32 g sin(θ)

Dividing both sides by g and rearranging for μ:

μ = (32/81) * (sin(θ)/cos(θ))

Using the angle θ = 30 degrees, where sin(30) = 0.5 and cos(30) = √3/2:

μ = (32/81) * (0.5 / (√3/2)) = (32/81) * (1/√3)

Calculating this gives:

μ = (32 / (81√3))

Final Result

Thus, the coefficient of friction for the rough inclined plane is:

μ ≈ 0.25 (approximately, depending on the numerical evaluation).

This result indicates the level of friction present on the rough incline compared to the smooth one, allowing us to understand how friction affects the motion of objects on inclined planes.