A high-speed test track for cars has a curved section — an arc of a circle of radius R = 3000 m. The curved section is banked at angle θ = 7 . 94 ◦ 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 side- ways. Answer in units of m / s
Sure, let's calculate the required speed v of the car.
To determine the speed, we need to consider the forces acting on the car when crossing the frictionless section. There are two main forces involved: the gravitational force (mg) and the normal force (N).
The gravitational force mg can be resolved into two components: mg sin(θ) and mg cos(θ), where θ is the banked angle.
The normal force N acts perpendicular to the incline and can be resolved into two components as well: N sin(θ) and N cos(θ).
At the point of crossing the frictionless section, the only horizontal force acting on the car is the component of the gravitational force pointing towards the center of the circle, mg sin(θ). This force provides the necessary centripetal force to keep the car moving in a circular path.
So, we can equate these forces:
mg sin(θ) = mv^2 / R
Where:
m = mass of the car
g = acceleration due to gravity
θ = banked angle
v = speed of the car
R = radius of the curved section
We can rearrange the equation to find the speed:
v = sqrt(g * R * tan(θ))
Now let's substitute the values:
g = 9.8 m/s^2 (acceleration due to gravity)
R = 3000 m (radius of the curved section)
θ = 7.94° (banked angle in radians: 7.94° x π / 180°)
v = sqrt(9.8 * 3000 * tan(7.94°))
Calculating this equation will give you the required speed v in meters per second (m/s).
To calculate the speed of the car that can cross the oil spill without slipping sideways, we need to consider the centripetal force and gravitational force acting on the car.
The centripetal force is provided by the normal force (N) and is given by:
F_c = m * a_c
where F_c is the centripetal force, m is the mass of the car, and a_c is the centripetal acceleration.
The centripetal acceleration is given by:
a_c = v^2 / R
where v is the speed of the car, and R is the radius of the curved section.
The gravitational force acting on the car can be decomposed into two components: one parallel to the banked surface (F_parallel) and one perpendicular to the banked surface (F_perpendicular).
The perpendicular component (F_perpendicular) is balanced by the normal force (N), given by:
N = m * g * sin(θ)
where g is the acceleration due to gravity, and θ is the angle of the banked surface.
The parallel component (F_parallel) provides the necessary centripetal force, and it is given by:
F_parallel = m * g * cos(θ)
Setting F_parallel equal to F_c, we have:
m * g * cos(θ) = m * v^2 / R
Dividing both sides of the equation by m and rearranging, we get:
g * cos(θ) = v^2 / R
Substituting the values given:
9.8 * cos(7.94°) = v^2 / 3000
Solving for v^2, we have:
v^2 = (9.8 * cos(7.94°)) * 3000
Taking the square root of both sides, we get:
v = √((9.8 * cos(7.94°)) * 3000)
Calculating this value, we find:
v ≈ 44.82 m/s
Therefore, the car needs to maintain a speed of approximately 44.82 m/s to cross the oil spill without slipping sideways.
To calculate the speed of a car that can cross the oil spill on the curved section without slipping sideways, we need to consider the forces acting on the car.
Here are the steps to solve the problem:
Step 1: Draw the free body diagram of the car on the banked curve.
The diagram will show the car on the banked curve with the forces acting on it. The forces involved are the normal force (N), the gravitational force (mg), the friction force (f), and the centripetal force (Fc).
Step 2: Calculate the normal force (N) and the gravitational force (mg).
The normal force (N) is the force exerted by the track perpendicular to the surface. In this case, it is the force that keeps the car from sinking into the inclined surface.
N = mg * cos(θ)
where m is the mass of the car and θ is the angle of inclination (7.94 degrees) from the horizontal.
Step 3: Calculate the friction force (f).
The friction force (f) is the force that prevents the car from slipping sideways. Since the section of the curved track is oil-spilled and frictionless, there is no friction force in that segment. Hence, the friction force will only act in the non-spilled section of the curve.
Step 4: Calculate the centripetal force (Fc).
The centripetal force (Fc) is the force that keeps the car moving along the curved section of the track.
Fc = m * (v^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 section (3000m).
Step 5: Equate the forces.
N - mg * sin(θ) = Fc
The net force in the vertical direction is the difference between the gravitational force component parallel to the inclined surface (mg * sin(θ)) and the component of the normal force perpendicular to the inclined surface (N).
Step 6: Solve for velocity (v).
Substituting the values and solving the equation, we get:
mg * cos(θ) - mg * sin(θ) = m * (v^2) / R
Rearrange the equation to solve for v:
v = sqrt((R * g * sin(θ))/(cos(θ) - sin(θ)))
Substitute the given values of g (9.8 m/s^2) and θ (7.94 degrees) to calculate v.
v ≈ 35.51 m/s
Therefore, the speed of the car that can cross the oil spill on the curve without slipping sideways is approximately 35.51 m/s.