Understanding Vertical Circular Motion
When an object is whirled in a vertical circle, the forces acting on the object change depending on its position. The primary forces involved are gravity (acting downwards) and the tension in the string (acting towards the center of the circle).
The Physics of Tension
For an object moving in a circle, the net force directed toward the center must equal the centripetal force:
$F_c = \frac{mv^2}{r}$
Where:
- $m$ is the mass of the object
- $v$ is the velocity
- $r$ is the radius
In a vertical circle, the tension ($T$) varies.
- At the bottom of the circle: The string must provide enough force to counteract gravity and provide the required centripetal force. Thus, the tension is at its maximum here: $T_{max} = \frac{mv^2}{r} + mg$.
- At the top of the circle: Gravity helps provide the centripetal force, so the tension is at its minimum: $T_{min} = \frac{mv^2}{r} - mg$.
Solving the Problem
Given:
- Mass ($m$) = $4\text{ kg}$
- Radius ($r$) = $1\text{ m}$
- Constant speed ($v$) = $3\text{ m/s}$
- Acceleration due to gravity ($g$) = $9.8\text{ m/s}^2$ (standard value)
Step 1: Calculate the centripetal force required. $F_c = \frac{mv^2}{r} = \frac{4 \times 3^2}{1} = \frac{4 \times 9}{1} = 36\text{ N}$
Step 2: Calculate the weight of the object. $W = mg = 4 \times 9.8 = 39.2\text{ N}$
Step 3: Determine the maximum tension. As established, the maximum tension occurs at the bottom of the circle: $T_{max} = F_c + W$ $T_{max} = 36\text{ N} + 39.2\text{ N} = 75.2\text{ N}$
Final Answer
The maximum tension in the string is 75.2 N.