A mass is moving in uniform circular motion under the effects of a centripetal force. If the force is doubled and the radius is cut in half, by what factor will the speed change?
It will double. <----
It will be cut in half.
It will not change.
It will quadruple.
It will be √2 times faster
F = m v^2/r so v^2 = F r/m
2F = m V^2/.5r so V^2 = F r/m
The same
The correct answer is:
It will not change.
In uniform circular motion, the speed is directly proportional to the radius of the circular path, and is given by the equation v = ωr, where v is the speed, ω is the angular velocity, and r is the radius of the circular path.
When the force is doubled, the centripetal force is given by F = mv^2/r, where m is the mass of the object. If the force is doubled, we have 2F = m(v')^2/r', where v' is the new speed, and r' is the new radius.
To find the factor by which the speed changes, we can compare the equations for the original and new situations. We have v^2 = ω^2r^2 and (v')^2 = ω^2(r')^2. Dividing the second equation by the first, we get (v')^2/v^2 = (r')^2/r^2. Since ω is the same for both situations, it cancels out.
Taking the square root of both sides, we have v'/v = r'/r. So, the ratio of the new speed to the original speed is equal to the ratio of the new radius to the original radius. Therefore, the speed will not change if the force is doubled and the radius is halved.
To answer this question, you can use the concept of centripetal force and circular motion. The centripetal force is given by the equation F = m * v^2 / r, where F is the force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.
In the given situation, the force is doubled while the radius is cut in half. Let's assume the original speed is v1 and the final speed is v2.
Initially, the centripetal force is F1 = m * v1^2 / r. After the changes are made, the new centripetal force becomes F2 = 2F1 = 2 * m * v1^2 / r1/2.
To find the final speed (v2), we need to equate the two forces:
2 * m * v1^2 / r1/2 = m * v2^2 / r
Now we can cancel out the mass (m) from both sides:
2 * v1^2 / r1/2 = v2^2 / r
Next, simplify the equation by substituting r1/2 as r/√2:
2 * v1^2 / (r / √2) = v2^2 / r
Now, rearrange the equation to solve for v2:
2 * v1^2 * √2 / r = v2^2
Take the square root of both sides:
√(2 * v1^2 * √2 / r) = v2
Simplify:
√(2 * v1^2 * √2) / √r = v2
Since √(2 * √2) is equal to 2, we can further simplify:
2 * v1 / √r = v2
Therefore, the final speed (v2) is equal to 2 times the initial speed (v1) divided by the square root of the radius (r).
So, the factor by which the speed changes is 2.