When a given object moves a circular path, the centripetal force varies directly as the square of the velocity and inversely as the radius of the curve. If the force is 640lb for a velocity of 20mi/h and a radius of 5m, find the force for the
velocity of 30mi/h and a radius of 4m.
F = kv^2/r
If
640 = k(20^2)/5, then the new force will be
640*(3/2)^2/(4/5) = 1800
To solve this problem, we'll use the concept of centripetal force and apply the given relationship.
According to the problem, the centripetal force (F) varies directly as the square of the velocity (v) and inversely as the radius (r). Mathematically, this can be expressed as:
F ∝ v^2 / r
Now, let's use the information provided in the problem to find the constant of variation. We're given that when the velocity is 20 mi/h and the radius is 5 m, the force is 640 lb.
Plugging these values into the equation, we get:
640 lb = k * (20 mi/h)^2 / 5 m
To simplify the equation, we need to convert the units consistently. Let's use the following conversions:
1 mi = 1609.34 m (approx.)
1 h = 3600 s (approx.)
So, we have:
640 lb = k * (20 * 1609.34 m/3600 s)^2 / 5 m
Simplifying further:
640 lb = k * (8.9408889 m/s)^2 / 5 m
640 lb = k * 15.9625581 m^2/s^2 / 5 m
640 lb = k * 3.19251162 m/s^2
Now, let's solve for the constant of variation (k):
k = 640 lb / 3.19251162 m/s^2
k ≈ 200.934943 lb * s^2/m
Now that we have the constant (k), we can use it to find the force for the given velocity of 30 mi/h and radius of 4 m.
Plugging the values into the equation, we get:
F = k * (30 mi/h)^2 / 4 m
Converting the units as before:
F = 200.934943 lb * s^2/m * (30 * 1609.34 m/3600 s)^2 / 4 m
Simplifying:
F = 200.934943 lb * s^2/m * (13.4114667 m/s)^2 / 4 m
F = 200.934943 lb * 179.73903235 m^2/s^2 / 4 m
F ≈ 900 lb
Therefore, the force for a velocity of 30 mi/h and a radius of 4 m is approximately 900 lb.