A 57.40 kg speed skater with a velocity of 12.10 m/s comes into a curve with a radius of 20.00 m. What is the minimum coefficient of friction for the skater to negotiate the curve?
To solve this problem, we need to consider the forces acting on the skater as they negotiate the curve. In this case, the only horizontal force is the friction force, which provides the centripetal force required to keep the skater moving in a curved path.
The centripetal force is given by the formula:
Fc = (m * v^2) / r
where Fc is the centripetal force, m is the mass of the skater, v is the velocity, and r is the radius of the curve.
In this case, the centripetal force is provided by the friction force:
Fc = f
where f is the friction force.
So we have:
f = (m * v^2) / r
Since we want to find the minimum coefficient of friction, we need to find the maximum value of f, which occurs when the skater is just about to slip.
The maximum value of the friction force is given by:
f_max = μ * N
where μ is the coefficient of friction and N is the normal force.
The normal force is equal to the weight of the skater, which is given by:
N = m * g
where g is the acceleration due to gravity.
Substituting the values into our equation, we have:
f_max = μ * m * g
Setting f_max equal to the centripetal force equation, we can solve for the minimum coefficient of friction:
(μ * m * g) = (m * v^2) / r
Simplifying the equation, we can solve for μ:
μ = (v^2) / (g * r)
Now we can substitute the given values into the equation to find the minimum coefficient of friction.