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 = (m * v^2) / r
where F is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.
However, in this problem, we are given the force, velocity, and radius, but we need to find the force for a different set of velocity and radius values.
Let's start by plugging in the given values into the formula:
640 lb = (m * (20 mi/h)^2) / 5 m
Now, we need to solve for the mass of the object (m).
Cross multiplying the equation gives us:
640 lb * 5 m = m * (20 mi/h)^2
Simplifying:
3200 lb * m = m * (400 mi^2/h^2)
Dividing both sides of the equation by m:
3200 lb = 400 mi^2/h^2
Now, we have determined that the mass of the object does not affect the resulting centripetal force, so we can ignore it in our calculations.
Next, let's use the new set of values to find the force:
F = (m * (30 mi/h)^2) / 4 m
Since we can ignore the mass of the object, the equation becomes:
F = (30 mi/h)^2 / 4 m
Simplifying:
F = 900 mi^2/h^2 / 4 m
Dividing:
F = 225 mi^2/h^2 / m
Now, since we know that the force does not depend on the mass of the object, we can use the same result we obtained earlier:
3200 lb = 400 mi^2/h^2
Therefore, the force for the velocity of 30 mi/h and a radius of 4 m is:
F = 225 mi^2/h^2 / m = 3200 lb / m = 3200 lb
Hence, the force is 3200 lb.