a body is launched up on a rough inclined plane making an angle of 30 degrees with yhe horizontal . obtain the coefficient of friction between body and the plane if the time of ascent is half of the time of descent
To obtain the coefficient of friction between the body and the inclined plane, we need to use the given information. Let's go step by step:
1. First, let's determine the time of ascent and time of descent. According to the given information, the time of ascent is half of the time of descent.
2. The time of ascent and descent can be determined using the formula for time of flight. When an object is launched vertically, the time of flight can be calculated using the equation:
t = 2*V₀*sinθ / g
Where:
t = time of flight
V₀ = initial velocity
θ = angle of the inclined plane with the horizontal
g = acceleration due to gravity (approximately 9.8 m/s²)
Since the time of ascent (t_a) is half of the time of descent (t_d), we can write:
t_a = 0.5*t_d
3. Now, let's find the expression for time of ascent and descent in terms of θ.
For time of ascent (t_a), we substitute V₀ with the upward component of velocity (V₀*sinθ):
t_a = 2*(V₀*sinθ) / g
For time of descent (t_d), we substitute V₀ with the downward component of velocity (V₀*cosθ):
t_d = 2*(V₀*cosθ) / g
4. Since we know that t_a = 0.5*t_d, we can substitute these expressions into the equation:
2*(V₀*sinθ) / g = 0.5 * (2*(V₀*cosθ) / g)
5. Simplifying this equation, we can cancel out the common factors:
sinθ = 0.5*cosθ
6. We can simplify further by dividing both sides by cosθ:
tanθ = 0.5
7. Taking the inverse tangent of both sides, we find the angle θ:
θ = arctan(0.5)
θ ≈ 0.464 radians
8. Finally, since we have the angle of inclination, we can use it to determine the coefficient of friction (μ) between the body and the inclined plane. The relationship between the coefficient of friction and the angle of inclination is given by the equation:
μ = tanθ
μ ≈ tan(0.464)
μ ≈ 0.508
Therefore, the coefficient of friction between the body and the plane is approximately 0.508.