The force needed to keep a car from skidding on a curve varies jointly as the weight of the car and the square of its speed and inversely as the radius of the curve. If 182lb of force keep a 1800lb car from skidding on a curve of radius 600ft at 30 mph, what force would keep the same car from skidding on a curve of radius 650ft at 45 mph?
The force needed to keep a car from skidding on a curve varies jointly as the weight of the car and the square of its speed and inversely as the radius of the curve. Given that it takes 2800 pounds of force to keep an 1800-pound car from skidding on a curve with a radius of 425 feet at 45 mph. We have to determine the force, rounded to the nearest 10 pounds that is needed to keep the same car from skidding when it takes a similar curve with a radius of 450 feet at 55 mph.
F = k * w * v^2 /R
182 = k (1800) (900)/600
k = .0674
F = .0674 (1800) (2025)/650
= 378 pounds
To solve this problem, we'll use the joint variation equation:
Force = k * (Weight) * (Speed^2) / (Radius)
where:
Force is the force needed to keep the car from skidding
Weight is the weight of the car
Speed is the speed of the car
Radius is the radius of the curve
k is the constant of variation
First, let's find the value of k using the given information. We'll use the first set of values provided: 182lb of force, 1800lb car weight, 600ft radius, and 30 mph speed.
182 = k * (1800) * (30^2) / (600)
To solve for k, we'll rearrange the equation:
k = 182 * (600) / (1800 * (30^2))
Now, we can use this value of k to find the force required for the second scenario, where the car weight is still 1800lb, the radius is 650ft, and the speed is 45mph.
Force = k * (1800) * (45^2) / (650)
Simplifying this equation, we can calculate the value of Force.
To find the force needed to keep the car from skidding on a curve of radius 650ft at 45 mph, we need to use the given information and apply the joint and inverse variation relationship.
Let's break down the given information:
Force needed to keep the car from skidding (F1) = 182 lb
Weight of the car (W) = 1800 lb
Radius of the curve (r1) = 600 ft
Speed of the car (s1) = 30 mph
Now, we need to find the force (F2) that would keep the same car from skidding on a curve of radius 650 ft at 45 mph.
Using the joint and inverse variation relationship, we can set up the following equation:
F1 = k * (W) * (s1)^2 / (r1)
where k is the constant of variation.
Rearranging the equation, we have:
k = F1 * (r1) / (W * (s1)^2)
Now, let's calculate the value of k:
k = (182 lb) * (600 ft) / (1800 lb * (30 mph)^2)
k ≈ 0.0404 lb * ft / (mph)^2
Now we can use the value of k to find the force F2:
F2 = k * (W) * (s2)^2 / (r2)
where
s2 = 45 mph (new speed)
r2 = 650 ft (new radius)
Substituting the values into the equation, we get:
F2 = (0.0404 lb * ft / (mph)^2) * (1800 lb) * (45 mph)^2 / (650 ft)
F2 ≈ 370 lb
Therefore, the force required to keep the same car from skidding on a curve of radius 650 ft at 45 mph is approximately 370 lb.