A high-speed test track for cars has a curved
section — an arc of a circle of radius R =
1440 m. The curved section is banked at
angle a = 19.1 degrees from the horizontal to help
the cars to stay in the road while moving at
high speeds.
The acceleration of gravity is 9.8 m/s^2 .
One day, oil spills on the track making a
few meters of the curved section frictionless.
Calculate the speed v of a car which can cross
the oil spill of the curve without slipping sideways.
Correct answer: 69.9051 m/s.
I'm supposed to figure out how they got this answer in order to apply this concept to a similar problem but I'm unsure as of how to do it.
To solve this problem, we need to consider the forces acting on the car as it passes through the frictionless section of the track. Since there is no friction, the only horizontal force acting on the car is the normal force from the banked road.
Let's break down the problem step by step:
1. Start by drawing a free body diagram of the car when it is passing through the frictionless section of the track.
- Draw the car with its weight acting vertically downward (mg), and the normal force (N) acting perpendicular to the curved road surface.
- Since there is no friction, there is no horizontal force acting on the car.
2. Resolve the weight into its components.
- The weight can be split into two components: one perpendicular to the road surface (mg sinθ) and one parallel to the road surface (mg cosθ).
- Here, θ refers to the angle of banking of the road (19.1 degrees).
3. Express the net force along the horizontal direction.
- Since the car does not slip sideways, the net force acting horizontally must be zero.
- The only horizontal force present is the component of the weight that is parallel to the road surface (mg cosθ).
- Therefore, the net force is given by:
Net force = mg cosθ.
4. Determine the centripetal force acting on the car.
- The centripetal force is the force that keeps the car moving in a circle.
- In this case, the centripetal force is provided by the horizontal component of the normal force (N cosθ).
- Therefore, the centripetal force is given by:
Centripetal force = N cosθ.
5. Relate the net force and the centripetal force.
- Since the net force and the centripetal force are both equal to mg cosθ, we can set them equal.
- Therefore: mg cosθ = N cosθ.
6. Calculate the normal force.
- We can solve for the normal force by substituting the known values into the equation:
mg cosθ = N cosθ.
- Since the mass of the car cancels out on both sides of the equation, we do not need the value of the car's mass.
- Therefore: N = mg.
Now that we have determined the normal force, we can calculate the speed of the car using the concept of centripetal force and circular motion.
7. Equate the centripetal force to the net force along the horizontal direction.
- The centripetal force acting on the car is equal to the net force, which is mg cosθ.
- Therefore, we can write: N cosθ = mg cosθ.
8. Express the centripetal force in terms of mass and velocity.
- The centripetal force acting on an object moving in a circular path is given by:
Centripetal force = (mass) x (velocity²) / (radius).
- In this case, the radius of the curved section is given as 1440 m.
9. Manipulate the equation to solve for the velocity.
- Substitute the expression for centripetal force into the equation:
N cosθ = (mass) x (velocity²) / (radius).
- Rearrange the equation to solve for the velocity:
velocity² = (N cosθ x radius) / (mass).
10. Calculate the velocity.
- Substitute the known values into the equation:
velocity² = (9.8 m/s² x 1440 m) / (9.8 m/s²).
- Simplify the equation:
velocity² = 1440 m².
- Take the square root of both sides to find the velocity:
velocity = sqrt(1440) m/s.
11. Calculate the final answer.
- Substitute the value of radius and solve:
velocity = sqrt(1440) m/s.
- The correct answer is approximately 69.9051 m/s.
By following these steps, you can calculate the required speed of the car to cross the oil spill on the curved section of the track without slipping sideways.