A 1200 kg car rounds a curve of radius 70m with a banking of 12°. The car is traveling 90 km/h, will a frictional force be required? If so, how much?
Well, if there's one thing I know about cars, it's that they always have a high demand for frictional forces, especially when it comes to rounding curves! They just can't get enough of that good ol' friction!
Now, let's get to the point. In order to determine whether a frictional force will be required, we need to examine the forces acting on the car. Remember, the banking angle of the curve plays a crucial role here.
If the banking angle is equal to or greater than the angle of friction, then no frictional force will be required. However, if the banking angle is less than the angle of friction, then a frictional force will indeed be needed.
So, assuming the curve is banked at 12°, and the coefficient of friction between the car's tires and the road is μ, we need to compare the two angles.
Now here's the fun part! I would love to calculate the exact value for you, but unfortunately, I need some information about the coefficient of friction (μ). Once provided, I'll be able to give you a more humorous response about whether or not a frictional force will be required and how much. So please, get that μ ready for me!
To determine if a frictional force is required for the car to safely navigate the curve, we need to compare the centripetal force required to keep the car moving in a circle with the maximum frictional force available between the tires of the car and the road.
Step 1: Convert the car's speed from km/h to m/s:
90 km/h = 90 * (1000 m / 1 km) / (3600 s / 1 h) = 25 m/s
Step 2: Calculate the centripetal force required using the formula:
F = m * v^2 / r
where F is the force, m is the mass, v is the velocity, and r is the radius.
F = (1200 kg) * (25 m/s)^2 / 70 m
F ≈ 13,095 N
Step 3: Calculate the maximum frictional force available using the formula:
F_friction = m * g * sin(θ)
where F_friction is the frictional force, m is the mass, g is the acceleration due to gravity, and θ is the banking angle.
θ = 12° = 12 * π / 180 radians
g = 9.8 m/s^2
F_friction = (1200 kg) * (9.8 m/s^2) * sin(12°)
F_friction ≈ 2,387 N
Step 4: Compare the required centripetal force with the maximum frictional force:
Since the required centripetal force (13,095 N) is greater than the maximum frictional force available (2,387 N), a frictional force is indeed required for the car to navigate the curve.
Step 5: Calculate the amount of required frictional force:
F_required = F - F_friction
F_required = 13,095 N - 2,387 N
F_required ≈ 10,708 N
Therefore, a frictional force of approximately 10,708 N is required for the car to safely navigate the curve.
To determine whether a frictional force will be required for the car to round the curve, we need to consider the forces acting on the car.
First, let's calculate the centripetal force required for the car to remain in a circular path. The centripetal force is provided by the normal force and can be calculated using the formula:
F_centripetal = m * v^2 / r
where
m = mass of the car = 1200 kg
v = velocity of the car = 90 km/h = 25 m/s
r = radius of the curve = 70 m
Plugging in these values, we get:
F_centripetal = 1200 kg * (25 m/s)^2 / 70 m = 10714.29 N
Next, we need to calculate the gravitational force acting on the car. The gravitational force can be calculated using the formula:
F_gravity = m * g
where
g = acceleration due to gravity = 9.8 m/s^2
Plugging in the values, we get:
F_gravity = 1200 kg * 9.8 m/s^2 = 11760 N
Now, let's calculate the angle of the incline (banking angle) using the formula:
θ = arctan(v^2 / (r * g))
Plugging in the values, we get:
θ = arctan((25 m/s)^2 / (70 m * 9.8 m/s^2)) = arctan(6.49) ≈ 77.21°
Comparing the banking angle (12°) with the angle of the incline (77.21°), we can see that the banking angle is less than the angle of the incline. Therefore, no frictional force is required because the gravitational force is sufficient to provide the necessary centripetal force.
In summary, a frictional force is not required for the car to round the curve.
v = 90000 m / 3600 s = 25 m/s
centripetal force necessary = 1200 * 25^2 / 70 = 10.7 kN
vertical force = m g = 1200 * 9.81 = 11.8 kN
normal force = m g cos(12º) = 11.5 kN
... horizontal component = 11.5 * sin(12º) = 2390 N
necessary horizontal friction force = 10.7 kN - 2.4 kN = 8.3 kN