A child slides down a slide with a 24o incline, and at the bottom her speed is precisely one-third what it would have been if the slide had been frictionless. Calculate the coefficient of kinetic friction between the slide and the child.
work done against friction = Initial PE - final Ke
= Force of friction * distance down slide
= mu * normal force * distance
Now where is your work?
To calculate the coefficient of kinetic friction between the slide and the child, we can use the given information about the inclination of the slide and the child's speed at the bottom.
Let's break down the problem step by step:
1. First, we need to understand the relationship between the child's speed and the incline of the slide. We can use trigonometry to find the component of gravitational force acting along the slide.
The gravitational force acting on the child can be split into two components: one along the incline and one perpendicular to it.
The component acting along the incline is given by: F_parallel = m * g * sin(theta), where m is the mass of the child, g is the acceleration due to gravity (approximately 9.8 m/s^2), and theta is the angle of inclination (24 degrees in this case).
2. The force of kinetic friction acting on the child is proportional to the component of gravitational force acting along the incline. We can express this relationship using the equation: F_friction = mu * F_normal, where mu is the coefficient of kinetic friction and F_normal is the normal force acting on the child.
3. To determine the normal force, we need to consider the forces acting on the child perpendicular to the incline. In this case, there are two forces: the component of gravitational force perpendicular to the incline (F_perpendicular = m * g * cos(theta)) and the normal force (F_normal).
Since the child is sliding down the slide and not lifting off it, the normal force is equal to the component of gravitational force perpendicular to the incline: F_normal = m * g * cos(theta).
4. Finally, the speed of the child at the bottom of the slide is related to the force of kinetic friction. According to the problem statement, the child's speed at the bottom is one-third of what it would be if the slide had been frictionless.
Using all the equations and information provided, we can solve for the coefficient of kinetic friction (mu):
F_parallel = F_friction
m * g * sin(theta) = mu * (m * g * cos(theta))
mu = (m * g * sin(theta)) / (m * g * cos(theta))
Simplifying the equation:
mu = tan(theta)
Now, plugging in the value of the angle of inclination (theta = 24 degrees), we can calculate the coefficient of kinetic friction:
mu = tan(24)
mu ≈ 0.445
Therefore, the coefficient of kinetic friction between the slide and the child is approximately 0.445.