A smooth block is released at an angle of 30 degree incline & then slide distance d. The time taken is 'x' times as much to slide on rough inclined plane than on smooth inclined. Then what is coefficient of friction?
To find the coefficient of friction, we need to analyze the situation and set up the necessary equations.
Let's consider the block sliding down a smooth inclined plane first:
1. In this case, there is no friction acting on the block, so the only force involved is the component of gravitational force pulling it down the slope.
2. The gravitational force can be broken down into two components: one parallel to the incline (mg*sinθ) and one perpendicular to the incline (mg*cosθ), where m is the mass of the block and θ is the angle of inclination.
3. The force parallel to the incline (mg*sinθ) is responsible for the acceleration of the block down the slope. Let's call it F_parallel_smooth.
4. According to Newton's second law, F_parallel_smooth = m * a_smooth, where a_smooth is the acceleration of the block.
5. Since the block is sliding down a smooth inclined plane, the acceleration can be calculated using the formula a_smooth = g*sinθ, where g is the acceleration due to gravity (9.8 m/s^2).
Now, let's consider the block sliding down a rough inclined plane:
1. In this case, there is friction acting on the block in addition to the gravitational force.
2. The force of friction (f_rough) can be calculated using the formula f_rough = μ * N, where μ is the coefficient of friction and N is the normal force.
3. The normal force acting on the block (N) can be calculated using the formula N = mg*cosθ.
4. The force parallel to the incline (mg*sinθ) minus the force of friction (f_rough) is responsible for the acceleration of the block down the slope. Let's call it F_parallel_rough.
5. According to Newton's second law, F_parallel_rough = m * a_rough, where a_rough is the acceleration of the block.
6. Since the time taken to slide on the rough inclined plane is 'x' times as much as on the smooth inclined plane, we can set up the equation:
t_rough = x * t_smooth
Since time = distance / velocity, we can rewrite the equation as:
d / v_rough = x * (d / v_smooth)
where v_rough is the velocity of the block on the rough incline and v_smooth is the velocity on the smooth incline.
7. The velocity of the block on a given incline can be calculated using the formula v = u + at, where u is the initial velocity, a is the acceleration, and t is the time.
Now, to find the coefficient of friction, we need to solve these equations simultaneously:
Equation 1: F_parallel_smooth = m * a_smooth
Equation 2: F_parallel_rough = m * a_rough
Equation 3: d / v_rough = x * (d / v_smooth)
By substituting the appropriate values and solving these equations, we can find the coefficient of friction (μ).
To find the coefficient of friction, we need to analyze the motion on the rough inclined plane and compare it to the motion on the smooth inclined plane.
Let's assume that the time taken to slide on the smooth inclined plane is T seconds.
1. Motion on the smooth inclined plane:
On the smooth inclined plane, the block moves with uniform acceleration along the incline. The acceleration can be calculated using the formula:
a = g * sin(θ), where g is the acceleration due to gravity and θ is the angle of incline.
2. Motion on the rough inclined plane:
On the rough inclined plane, there is an additional force acting against the motion due to friction. This force opposes the motion and reduces the acceleration.
Now, according to the given information, the time taken to slide on the rough inclined plane is 'x' times as much as the time taken on the smooth inclined plane. Therefore, the time taken on the rough inclined plane is xT seconds.
Since the time taken is directly proportional to the square root of the distance traveled, we can write the following equation:
sqrt(d) = x * sqrt(m),
where d is the distance traveled on the rough inclined plane and m is the distance traveled on the smooth inclined plane.
3. Relationship between distance and time:
The distance traveled can be calculated using the equation of motion:
d = v * t + (1/2) * a * t^2,
where v is the initial velocity, t is the time taken, and a is the acceleration.
Since the block is released, the initial velocity on both planes is zero.
On the smooth inclined plane:
d = (1/2) * (g * sin(θ)) * T^2
On the rough inclined plane:
d = (1/2) * (g * sin(θ) - μ * g * cos(θ)) * (xT)^2,
where μ is the coefficient of friction.
4. Solve for the coefficient of friction:
Now, let's substitute the values of d on both planes and solve for μ.
(1/2) * (g * sin(θ)) * T^2 = (1/2) * (g * sin(θ) - μ * g * cos(θ)) * (xT)^2
Canceling out common factors and rearranging the terms:
(sin(θ))/T^2 = (sin(θ) - μ * cos(θ))/ (x^2 * T^2)
Dividing both sides by sin(θ):
1/T^2 = (1 - (μ/sin(θ)) * cos(θ))/(x^2 * T^2 * sin(θ))
Simplifying further:
1 = (1 - (μ/sin(θ)) * cos(θ))/ (x^2 * sin(θ))
Now, solving for μ:
μ/sin(θ) = 1 - 1/x^2
μ = sin(θ) - sin(θ)/x^2
Therefore, the coefficient of friction is given by:
μ = sin(θ) * (1 - 1/x^2)