Understanding Motion in a Vertical Circle
When an object is whirled in a vertical circle, the forces acting on it—specifically gravity and the tension of the string—change as the object moves from the top to the bottom.
The Physics of Tension
At any point in a vertical circular path, the net force directed toward the center of the circle provides the centripetal force required to keep the object moving in a circle. The centripetal force is given by: $$F_c = \frac{mv^2}{r}$$
Where:
- $m$ is the mass of the object
- $v$ is the constant speed
- $r$ is the radius of the circle
Where is the Tension Maximum?
At the bottom of the circle, two forces act on the object:
- Tension ($T$): Acts upward (toward the center).
- Weight ($mg$): Acts downward (away from the center).
The net force toward the center is $T - mg$. Setting this equal to the centripetal force: $$T - mg = \frac{mv^2}{r}$$ $$T = \frac{mv^2}{r} + mg$$
This is why tension is at its maximum at the bottom: the string must support both the object's weight and provide the necessary centripetal force.
Step-by-Step Solution
Given:
- Mass ($m$) = $4\text{ kg}$
- Radius ($r$) = $1\text{ m}$
- Speed ($v$) = $3\text{ m/s}$
- Gravity ($g$) = $9.8\text{ m/s}^2$ (standard approximation)
Calculation:
- Calculate Centripetal Force ($F_c$): $$F_c = \frac{4 \times 3^2}{1} = 4 \times 9 = 36\text{ N}$$
- Calculate Weight ($W$): $$W = mg = 4 \times 9.8 = 39.2\text{ N}$$
- Calculate Max Tension ($T_{max}$): $$T_{max} = F_c + W = 36 + 39.2 = 75.2\text{ N}$$
Result: The maximum tension in the string is 75.2 N.