What effect does doubling the radius of the circle that an object travels have on its centripetal force, if everything else is constant?
I think it should double centripetal force, but that is marked wrong, even though I used a test case. My test case was 22kg mass, 1.25-2.5m radius, and 4.19 rad/sec angular velocity.
Why is my answer wrong?
cf = m v^2 / r
v is tangential velocity ... not angular
doubling r halves the cf
... with everything else constant
BUT v = omega R
I assume it is omega, not v that is constant here
another way to look at it is
Ac = centripetal acceleration
= omega^2 R
where omega is the angular velocity which does not change in this example.
then
Ac = omega^2 * 2 R
And I think your answer was correct
yes, "everything else" is certainly ambiguous
but the answer was marked wrong
the originator of the question will have the final say
good luck
The issue is "everything else"..does that include angular velocity, or velocity? Your teacher was very sloppy in writing this question. I hope that is not a common occurance.
Your answer is incorrect because the centripetal force of an object traveling in a circle depends on the square of the radius, not the radius itself. Doubling the radius of the circle means increasing it by a factor of 2, and according to the formula for centripetal force, the force is proportional to the square of the radius.
To explain this further, the centripetal force (Fc) is given by the formula:
Fc = (m * v^2) / r
where m is the mass of the object, v is the velocity/linear speed, and r is the radius of the circle.
In your example, you mentioned a mass of 22 kg, a radius ranging from 1.25 m to 2.5 m, and an angular velocity of 4.19 rad/sec. Let's consider the initial case where the radius is 1.25 m. To calculate the centripetal force, we need to find the velocity (v) using the angular velocity (ω) and the radius (r):
v = ω * r
v = 4.19 rad/sec * 1.25 m
v ≈ 5.238 m/s
Now let's calculate the centripetal force using the initial radius:
Fc1 = (22 kg * (5.238 m/s)^2) / 1.25 m
Fc1 ≈ 92.018 N
Now, if we double the radius to 2.5 m, we can calculate the new centripetal force:
Fc2 = (22 kg * (5.238 m/s)^2) / 2.5 m
Fc2 ≈ 46.009 N
As you can see, doubling the radius doesn't double the centripetal force. It actually reduces the centripetal force by a factor of 2, since the force is inversely proportional to the square of the radius.