Question

A small mass m is tied to a string of length L and is whirled in vertical circular motion. The speed of the mass v is such that the ratio of the string tension at the top of the circle to that at the bottom of the circle is FtopT/FbotT = 0.5. Derive an expression for the speed v.

Answer 1

To derive an expression for the speed v in a vertical circular motion, we can use the concept of centripetal force. By equating the tension at the top and bottom of the circle and using the given ratio, we can solve for the speed v. The expression for the speed v is v = sqrt(29.4L).

Step-by-step explanation:

To derive an expression for the speed v, we can use the concept of centripetal force. At the top of the circular motion, the tension T is the sum of the mass m multiplied by g (the acceleration due to gravity) and the centripetal force. At the bottom of the circular motion, the tension T is the difference between the mass m multiplied by g and the centripetal force. By equating these two tensions and using the given ratio of FtopT/FbotT = 0.5, we can solve for the speed v.

Let's proceed with the derivation:

1. For the top of the circle: Ttop = mg + mv2/L
2. For the bottom of the circle: Tbot = mg - mv2/L
3. Equating the two tensions: mg + mv2/L = 0.5(mg - mv2/L)
4. Simplifying the equation: 2mg + 2mv2/L = mg - mv2/L
5. Further simplifying: 2mg = (mg - mv2/L) * 0.5
6. Expanding and rearranging the equation: 4mg = mg - mv2/L
7. Simplifying: 3mg = mv2/L
8. Substituting the value of g: 3m(9.8 m/s2) = mv2/L
9. Simplifying: v2 = 29.4L
10. Taking the square root of both sides: v = sqrt(29.4L)

Therefore, the expression for the speed v is v = sqrt(29.4L).