Animated Solution for Physics - Properties of Solids and Liquids: A body of mass $m=10\text{ kg}$ is attached to one end of a wire of length $0.3\text{ m}$. The maximum angular speed (in ${\text{rad s}}^{-1}$) with which it can be rotated about its other end in space station is (breaking stress of wire $=4.8\times 10^7{\text{ Nm}}^{-2}$ and area of cross-section of the wire $=10^{-2}{\text{ cm}}^2$) is

Imagine you are floating in a space station, completely free from the relentless pull of Earth's gravity. You have a mass of $10\text{ kg}$ attached to a wire of length $0.3\text{ m}$, and you decide to spin it around. Because you are in a microgravity environment, the only force acting on the mass to keep it moving in a circle is the tension in the wire. There is no $mg$ to worry about!

For any object to move in a circular path, it requires a centripetal force directed towards the center of the circle. In our setup, this crucial force is provided entirely by the tension $T$ in the wire.

Mathematically, we express this as:

Here, $m$ is the mass, $\omega$ is the angular speed, and $l$ is the length of the wire (which acts as the radius of the circular path). As you spin the mass faster, the angular speed $\omega$ increases, and consequently, the tension $T$ in the wire shoots up.

Every material has a limit. As the wire pulls the mass, it experiences stress, which is defined as the internal restoring force per unit area.

Substituting our expression for tension, the stress in the wire becomes:

The wire will snap if this stress exceeds its maximum capacity, known as the breaking stress (${\sigma }_{\text{max}}$). To find the absolute maximum angular speed ${\omega }_{\text{max}}$ before disaster strikes, we set the stress exactly equal to the breaking stress:

Now, we must be incredibly careful with our units. A classic trap is forgetting to convert the cross-sectional area into standard SI units.

The area is given as $10^{-2}{\text{ cm}}^2$. Since $1\text{ cm}=10^{-2}\text{ m}$, squaring it gives $1{\text{ cm}}^2=10^{-4}{\text{ m}}^2$. Therefore:

Let's plug all our values into the master equation:

The numerator simplifies beautifully: $4.8\times 10^1=48$. The denominator is simply $3$.

Taking the square root of both sides, we arrive at our final answer:

If you spin the mass even a fraction faster than $4\text{ rad/s}$, the stress will exceed $4.8\times 10^7{\text{ N/m}}^2$, and the wire will break. It is a beautiful demonstration of how material properties dictate the limits of dynamic motion!