What is the maximum speed for a car to safely negotiate a 30 banked curve of 60 m radius if there is a coefficient of static friction of 0.35?
The weight of the car parallel to the bank: mgCosTheta
friction : mg*Mu*sinTheta
centrepetal force up the bank
mv^2/r*cosTheta
Normal force component of centripetal force adding to weight:
mv^2/r*mu*SinTheta.
forces up the bank=forces down
friction+weightdownbank=centripal force up
mv^2*mu*Sintheta+mg*mu*CosTheta+mg*sinTheta=mv^2/r* CosTheta
solve for v.
To determine the maximum speed for a car to safely negotiate a banked curve, we need to consider the centripetal force and the frictional force acting on the car. The centripetal force acts towards the center of the curved path and is provided by the horizontal component of the car's weight, while the frictional force acts towards the center of the curve, opposing the car's tendency to slide outward.
The maximum speed occurs when the frictional force reaches its maximum value. This occurs when the static friction equals the product of the coefficient of static friction and the normal force (N), which is the component of the car's weight perpendicular to the surface.
To begin, we'll calculate the normal force (N) acting on the car. The weight of the car (W) is given by the formula:
W = mg
where m is the mass of the car, and g is the acceleration due to gravity. However, in this case, we are interested in the normal force (N), which is the component of the weight perpendicular to the curved surface. Therefore, the normal force can be calculated using:
N = W cos(θ)
where θ is the angle of the banked curve (30 degrees).
Next, we will calculate the centripetal force (Fc) required to keep the car moving in a circular path. The centripetal force is given by the formula:
Fc = (mv^2) / r
where m is the mass of the car, v is the velocity of the car, and r is the radius of the curved path.
Since the centripetal force is provided by the horizontal component of the car's weight, we have:
Fc = W sin(θ)
Now, we can equate the frictional force (Ff) to the centripetal force (Fc) to find the maximum speed safely negotiated around the banked curve:
Ff = μN = μW cos(θ)
Fc = W sin(θ)
Setting these equal, we get:
μW cos(θ) = W sin(θ)
We can cancel out the weight (W) from both sides of the equation:
μ cos(θ) = sin(θ)
Now, we can solve for θ by taking the inverse tangent of both sides:
θ = arctan(μ)
In this case, the given coefficient of static friction (μ) is 0.35. So, we can calculate the angle θ:
θ = arctan(0.35) ≈ 19.28 degrees
Finally, we have all the necessary information to determine the maximum speed (v) at which the car can safely negotiate the banked curve. Using the equation for the centripetal force (Fc), we can rearrange it to solve for v:
v = √(r * g * tan(θ))
Substituting the known values, we can calculate the maximum speed:
v = √(60 * 9.8 * tan(19.28))
v ≈ 20.6 m/s
Therefore, the maximum speed for the car to safely negotiate the 30-degree banked curve with a radius of 60 meters and a coefficient of static friction of 0.35 is approximately 20.6 meters per second.