A car weighing 3220 lb a curve of 200 ft radius bank at an angle of 30°. Find the friction force acting on tires when the car is traveling at 60 mph. The coefficient of friction between the tires and the road is 0.9.
show solution step by step
show FBD (if possible)
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To find the friction force acting on the tires, we can start by considering the forces acting on the car when it is traveling on the curved road.
Step 1: Free Body Diagram (FBD):
Draw a free body diagram to understand the forces acting on the car. In this case, the forces acting on the car are:
- Gravity acting vertically downward with a magnitude equal to the weight of the car (3220 lb or 3220 * 32.2 lb-ft/s^2). Label it as "W".
- Normal force (N) acting perpendicular to the road surface.
- Friction force (Ff) acting in the opposite direction of the car's motion.
Step 2: Resolve the forces:
Resolve the forces acting on the car into horizontal and vertical components.
- Resolve the weight of the car (W) into its horizontal and vertical components. The vertical component is equal to W * cos(30°). Label it as "Wv". The horizontal component is equal to W * sin(30°) and acts towards the center of the curve. Label it as "Wh".
- The normal force (N) is perpendicular to the road surface and acts in the vertical direction, balancing the weight of the car. Therefore, N = Wv.
Step 3: Find the friction force:
The friction force is equal to the horizontal component of the weight, acting towards the center of the curve, which can be calculated using the equation:
Ff = μ * N,
where μ is the coefficient of friction between the tires and the road.
- Substitute the value of N as Wv.
- Given that the coefficient of friction is 0.9, substitute this value into the equation.
Step 4: Convert the car's weight and radius of the curve:
- Convert the weight of the car from pounds to Newtons by multiplying by the conversion factor 32.2 lb-ft/s^2 = 1 lb (force).
- Convert the radius of the curve from feet to meters, as the SI unit is preferred. Multiply the given radius (200 ft) by 0.3048 m/ft.
Step 5: Calculate the velocity of the car:
- Convert the given velocity of the car from mph to m/s by multiplying by the conversion factor 0.447 m/s = 1 mph.
Step 6: Substitute the known values and solve for the friction force:
- Substitute the known values into the equation Ff = μ * N and solve for Ff.
Step 7: Calculate the friction force:
- Evaluate the numerical value of Ff.
By following these steps, you can calculate the friction force acting on the tires when the car is traveling at 60 mph around a curve of 200 ft radius banked at an angle of 30°.
To find the friction force acting on the tires, we need to consider the forces acting on the car such as the gravitational force and the centripetal force.
Step 1: Calculate the centripetal force.
The centripetal force required for the car to make a curve is given by the formula:
F = m * v^2 / r
where F is the centripetal force, m is the mass of the car, v is the velocity, and r is the radius of the curve.
First, convert the weight of the car from pounds to slugs (since the formula requires the mass in slugs):
Weight in slugs = weight in pounds / 32.2
= 3220 lb / 32.2
= 100 slugs
Next, convert the velocity from miles per hour to feet per second:
Velocity in ft/s = velocity in mph * 1.47
= 60 mph * 1.47
= 88 ft/s
Now, calculate the centripetal force:
F = (100 slugs) * (88 ft/s)^2 / 200 ft
≈ 2,432 lb
Step 2: Calculate the maximum friction force.
The maximum friction force can be calculated using the equation:
Friction force = coefficient of friction * normal force
where the normal force is the vertical component of the weight of the car.
Normal force = weight of the car * cos(angle of bank)
= 3220 lb * cos(30°)
= 3220 lb * (√3 / 2)
≈ 3220 lb * 0.866
≈ 2790 lb
Now calculate the maximum friction force:
Friction force = 0.9 * 2790 lb
≈ 2,511 lb
Step 3: Determine the friction force.
Since the centripetal force (2,432 lb) is less than the maximum friction force (2,511 lb), the friction force is equal to the centripetal force.
So the friction force acting on the tires is approximately 2,432 lb.
FBD (Free Body Diagram):
The FBD of the car on the banked curve would include the gravitational force acting vertically downwards, the normal force perpendicular to the surface of the road, and the friction force acting horizontally towards the center of the curve.
-> F (friction force)
_↑_
|__|
weight -> |__________|
I hope this helps! Let me know if you have any further questions.