Sure, I'd be happy to help you understand the steps to solve this problem.
To find the maximum speed (Vmax) that a car can go around the curve without sliding out, we need to consider the forces acting on the car. There are two main forces involved: the gravitational force (mg) and the frictional force (Ff) between the tires and the road.
Step 1: Identify the forces:
The gravitational force (mg) acts vertically downwards and can be represented as mg = m * g, where m is the mass of the car and g is the acceleration due to gravity.
Step 2: Resolve the gravitational force:
Resolve the gravitational force into two components: one parallel to the surface of the road (mg * sinθ) and one perpendicular to the surface of the road (mg * cosθ), where θ is the bank angle.
Step 3: Determine the frictional force:
The frictional force (Ff) opposes the tendency of the car to slide out of the curve. It acts towards the center of the circle and can be represented as Ff = μs * (mg * cosθ), where μs is the coefficient of static friction.
Step 4: Equate the forces:
The net force acting on the car is the difference between the frictional force and the component of gravitational force parallel to the road. Equate these two forces to find the maximum speed:
Ff = mg * sinθ
μs * (mg * cosθ) = mg * sinθ
Step 5: Cancel out common terms:
Cancel out the mass (m) from both sides of the equation.
μs * cosθ = sinθ
Step 6: Solve for the maximum speed:
Rearrange the equation to solve for Vmax:
Vmax = √(R * g * tanθ)
Step 7: Plug in the values and convert units:
Substitute the given values into the equation and calculate Vmax.
Vmax = √(R * g * tanθ)
Vmax = √(174 * 9.8 * tan(14°))
Finally, convert the answer to km/hr by multiplying it by (60 * 60) / 1000 to convert from m/s to km/hr.
I hope this step-by-step explanation helps you understand how to arrive at the solution. Let me know if you have any further questions!