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.

To solve this problem, we can use the formula for centripetal force:

F = (mv²) / r,

where F is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular path.

We are given the force, velocity, and radius for the first set of conditions:

F₁ = 640 lb,
v₁ = 20 mi/h,
r₁ = 5 m.

We need to find the force for the second set of conditions:

v₂ = 30 mi/h,
r₂ = 4 m.

We can set up a proportion using the given information:

F₁ / (v₁² * r₁) = F₂ / (v₂² * r₂),

where F₂ represents the unknown force.

Now let's substitute the given values into the proportion:

640 lb / (20 mi/h)² * 5 m = F₂ / (30 mi/h)² * 4 m.

Simplifying the equation:

640 / (400) * 5 = F₂ / (900) * 4.

Solving for F₂:

F₂ = (640 * 900 * 5) / (400 * 4).

F₂ ≈ 2025 lb.

Therefore, the force for a velocity of 30 mi/h and a radius of 4 m is approximately 2025 lb.