A body of mass “m” has an initial velocity “v0” directed up a plane that is at an inclination angle to the
horizontal. The coefficient of sliding friction between the mass and the plane is μ. What distance “s” will
the body slide up the plane before coming to rest?
To find the distance "s" that the body will slide up the plane before coming to rest, we can use the equations of motion and the principles of Newton's laws.
Let's break down the problem step by step:
1. First, determine the forces acting on the body. In this case, we have the force of gravity acting vertically downwards and the force of friction acting up the inclined plane. The force of gravity can be calculated using the formula Fg = m * g, where m is the mass of the body and g is the acceleration due to gravity.
2. The force of friction can be calculated using the formula Ff = μ * N, where μ is the coefficient of sliding friction and N is the normal force. The normal force is the component of the gravitational force acting perpendicular to the inclined plane and can be calculated as N = m * g * cos(θ), where θ is the inclination angle.
3. Next, calculate the net force acting on the body along the inclined plane. The net force is given by Fnet = m * a, where a is the acceleration of the body. Since the body is coming to rest, the net force is zero. Therefore, we have Fnet = Ff - Fg = 0.
4. Substituting the values of Ff and Fg, we get μ * N - m * g = 0.
5. Rearranging the equation, we can solve for the normal force N: N = m * g * cos(θ).
6. Now, substitute the value of N into the previous equation to find the value of μ * m * g * cos(θ) - m * g = 0.
7. Simplifying the equation further, we find μ * cos(θ) - 1 = 0.
8. Solve for the value of cos(θ) by dividing both sides by μ: cos(θ) = 1 / μ.
9. Take the inverse cosine of both sides to find the value of θ: θ = arccos(1 / μ).
10. Finally, calculate the distance "s" using the formula s = (v0^2) / (2 * g * sin(2θ)), where v0 is the initial velocity of the body directed up the plane (perpendicular to the horizontal) and g is the acceleration due to gravity.
By following these steps and substituting the known values into the equations, you can determine the distance "s" that the body will slide up the plane before coming to rest.