A tennis ball on a string is being swung in a horizontal circle of radius r. The ball has mass m and the string length l. Which expression gives the magnitude of the tension T in the string in terms only of the speed v of the ball and of m, r, and l?

Answer choices
1. T=mgr/l
2. T=0, as is always the case for a conical pendulum.
3. T=mv^2/l
4. T=mv^2/r
5. T=m[g−(v^2/r)]
6. T=mv^2/2
7. T=mg
8. T=m[g+(v^2/r)]
9. T=mv^2l/r^2

Actual Answer is 9

SA's answer of 4 is false, I tried it out personally, it doesn't work

I do not see where the length of the string (l) comes in the picture if the radius of the circular motion is fixed at r.

If the ball is swung in air where it is allowed to sag, then there is the horizontal component equal to mv²/r and the vertical component equal to mg.
The vectorial sum is
m*sqrt(g²+(v²/r)²)
which does not correspond to any of the options listed, unless the square-root sign (√) is missing from choices 5 and 8.

The correct expression for the magnitude of the tension T in the string can be found using the centripetal force equation. The centripetal force is given by the formula F = mv^2/r, where m is the mass of the ball, v is the speed of the ball, and r is the radius of the circle.

However, in this case, the tension T in the string is responsible for providing the centripetal force. So, we can set T equal to mv^2/r. Therefore, the correct expression for T in terms of m, v, and r is:

T = mv^2/r

The correct answer is 4. T=mv^2/r.

To find the expression that gives the magnitude of the tension T in the string, we can consider the forces acting on the tennis ball when it is swung in a horizontal circle.

First, let's analyze the forces acting on the ball:
1. The weight of the ball, which is mg, acts vertically downward.
2. The tension T in the string, which acts towards the center of the circle.
3. The centripetal force, which is mv^2/r, acts towards the center of the circle.

Since the ball is not moving up or down (vertically), the vertical forces must be balanced. Therefore, the tension T must be equal to the weight of the ball, mg.

Now, let's look at the answer choices:

1. T = mgr/l: This expression doesn't take into account the horizontal motion and the centripetal force. Incorrect.
2. T = 0: This answer choice assumes that the tension T is always zero for a conical pendulum. However, the given scenario describes a horizontal circular motion, not a conical pendulum. Incorrect.
3. T = mv^2/l: This expression only considers the centripetal force and doesn't account for the weight of the ball. Incorrect.
4. T = mv^2/r: This is the correct expression for the tension T in the string. It considers both the centripetal force and the weight of the ball. The centripetal force mv^2/r is required to keep the ball moving in a circle, and the weight mg provides the downward force. Therefore, the magnitude of the tension T is equal to mv^2/r. This is the correct answer.
5. T = m[g - (v^2/r)]: This expression considers the weight of the ball and the centripetal force, but the signs are incorrect. The weight mg and the centripetal force mv^2/r both act towards the center of the circle, so they should have the same sign. Incorrect.
6. T = mv^2/2: This expression doesn't take into account the radius of the circle or the weight of the ball. Incorrect.
7. T = mg: This only considers the weight of the ball and doesn't account for the horizontal motion and the centripetal force. Incorrect.
8. T = m[g + (v^2/r)]: This expression considers the weight of the ball and the centripetal force, but the signs are incorrect. The weight mg and the centripetal force mv^2/r both act towards the center of the circle, so they should have the same sign. Incorrect.
9. T = mv^2l/r^2: This expression includes the length of the string in the denominator, which is incorrect. The tension in the string is mainly determined by the centripetal force and the weight of the ball, not the length of the string. Incorrect.

Therefore, the correct expression for the magnitude of the tension T in the string is T = mv^2/r, which is answer choice 4.