A 38 kg child is sitting on a slide at angle 45o from the horizontal. What is the minimum value of static friction between the child and the slide that prevents the child from sliding down?
To determine the minimum value of static friction that prevents the child from sliding down the slide, we need to consider the forces acting on the child. In this case, there are two forces involved: the gravitational force pulling the child downward and the static friction force opposing the child's motion.
We can break down the gravitational force into its components. The component parallel to the slide is responsible for trying to make the child slide down, while the component perpendicular to the slide does not affect the child's slide.
The parallel component of the gravitational force can be calculated using the formula:
Force_parallel = Mass * Acceleration_due_to_gravity * sin(θ)
where Mass is the mass of the child (38 kg), Acceleration_due_to_gravity is the acceleration due to gravity (9.8 m/s^2), and θ is the angle of the slide relative to the horizontal (45°).
Substituting the given values:
Force_parallel = 38 kg * 9.8 m/s^2 * sin(45°)
Next, we need to find the maximum static friction force, as this determines the minimum value of static friction that prevents sliding. In this case, it's equal to the force preventing the child from sliding down the slide, which is the parallel component of the gravitational force. So:
Maximum_static_friction_force = Force_parallel
Now, we can calculate the minimum value of static friction using the formula:
Minimum_static_friction_force = Coefficient_of_static_friction * Normal_force
where Coefficient_of_static_friction is the coefficient of static friction between the child and the slide, and Normal_force is the perpendicular force exerted by the slide on the child. In this case, the normal force is equal to the component of the gravitational force perpendicular to the slide surface:
Normal_force = Mass * Acceleration_due_to_gravity * cos(θ)
Substituting the given values:
Normal_force = 38 kg * 9.8 m/s^2 * cos(45°)
Finally, to find the minimum value of static friction, we equate the maximum static friction force to the minimum static friction force:
Force_parallel = Coefficient_of_static_friction * Normal_force
Substituting the previously calculated values, we can solve for the coefficient of static friction:
38 kg * 9.8 m/s^2 * sin(45°) = Coefficient_of_static_friction * 38 kg * 9.8 m/s^2 * cos(45°)
By simplifying and canceling out common terms, we get:
sin(45°) = Coefficient_of_static_friction * cos(45°)
Now, we can solve for the coefficient of static friction:
Coefficient_of_static_friction = sin(45°) / cos(45°)
Evaluating this expression:
Coefficient_of_static_friction = 1
Therefore, the minimum value of static friction between the child and the slide that prevents the child from sliding down is 1.