Suppose you want to negotiate a curve with a radius of 50 meters and a bank angle of 15. If the coefficient of friction between your tires and the pavement is 0.50, what is the maximum speed that you can safely use? (NB: friction acts down the slope).
To find the maximum speed that you can safely use to negotiate the curve, we need to consider the forces acting on the vehicle and determine the point where the maximum frictional force can balance the centripetal force.
The centripetal force required to negotiate the curve is given by:
F_c = m * v^2 / r
where:
F_c is the centripetal force,
m is the mass of the vehicle,
v is the velocity of the vehicle, and
r is the radius of the curve.
In this case, we need to consider the direction of the frictional force, which acts down the slope due to the banking angle. The maximum frictional force that can be exerted is given by:
F_friction = μ * m * g
where:
F_friction is the maximum frictional force,
μ is the coefficient of friction between the tires and pavement,
m is the mass of the vehicle, and
g is the acceleration due to gravity.
To determine the maximum speed, we equate the centripetal force to the maximum frictional force:
F_c = F_friction
m * v^2 / r = μ * m * g
Notice that the mass cancels out and we can solve for v:
v^2 = μ * r * g
v = √(μ * r * g)
Now, we can substitute the given values:
μ = 0.50 (coefficient of friction)
r = 50 meters (radius of the curve)
g ≈ 9.81 m/s^2 (acceleration due to gravity)
Plugging these values into the equation, we get:
v = √(0.50 * 50 * 9.81)
Calculating the value:
v ≈ √(245.25)
v ≈ 15.65 m/s
Therefore, the maximum speed that you can safely use to negotiate the curve is approximately 15.65 m/s.