Question

A sphere of mass 20 kg is suspended by a metal wire of unstretched length 4 m and diameter 1 mm. When in equilibrium, there is a clear gap of 2 mm between the sphere and the floor. The sphere is gently pushed aside so that the wire makes an angle $\theta$ with the vertical and is released. Find the maximum value of $\theta$ so that the sphere does not rub the floor. Young's modulus of the metal of the wire is $2.0 \times 10^{11} N/m^2$. Make appropriate approximations.

Answer 1

The problem as stated is ill-posed, as the minimum height of the sphere above the floor is $2$ mm, meaning it never rubs the floor. If interpreted as the sphere just touching the floor at maximum displacement, it leads to an impossible physical scenario ($\cos \theta > 1$).

Answer 2

The problem describes a sphere suspended by a wire, forming a pendulum. In equilibrium, the wire has an unstretched length $L_0 = 4$ m and an extension $\Delta L_0$ due to the weight of the sphere. The gap between the sphere and the floor in equilibrium is $h_{floor} = 2$ mm.

1. Calculate Equilibrium Extension ($\Delta L_0$):

   • Mass of sphere, $m = 20$ kg
   • Young's modulus, $Y = 2.0 \times 10^{11}$ N/m$^2$
   • Diameter, $d = 1$ mm $= 1 \times 10^{-3}$ m $\implies$ Radius, $r = 0.5 \times 10^{-3}$ m
   • Cross-sectional Area, $A = \pi r^2 = \pi (0.5 \times 10^{-3})^2 = 0.25\pi \times 10^{-6}$ m$^2$
   • Tension in equilibrium, $T_0 = mg = 20 \times 9.8 \approx 200$ N (using $g \approx 9.8$ m/s$^2$ or $10$ m/s$^2$). Let's use $g \approx 10$ m/s$^2$ for simplicity as approximations are allowed. $T_0 = 200$ N.
   • Extension: $\Delta L_0 = \frac{T_0 L_0}{A Y} = \frac{(200 \text{ N})(4 \text{ m})}{(0.25\pi \times 10^{-6} \text{ m}^2)(2.0 \times 10^{11} \text{ N/m}^2)} = \frac{800}{0.5\pi \times 10^5} = \frac{1600}{\pi} \times 10^{-5}$ m.
   • Using $\pi \approx 3.14$, $\Delta L_0 \approx \frac{1600}{3.14} \times 10^{-5} \approx 509.6 \times 10^{-5}$ m $\approx 5.1$ mm.

2. Analyze Sphere's Height:

   • The equilibrium length of the wire is $L_{eq} = L_0 + \Delta L_0 = 4 \text{ m} + 5.1 \text{ mm} \approx 4.0051$ m.
   • Let the suspension point be O. The distance from O to the floor is $H_{total} = L_{eq} + h_{floor} = (4.0051 \text{ m}) + (2 \times 10^{-3} \text{ m}) \approx 4.0071$ m.
   • When the wire makes an angle $\phi$ with the vertical, the length of the wire is approximately $L_{eq}$ (since $\Delta L_0 \ll L_0$, the change in length due to tension variation is negligible for this approximation).
   • The vertical distance from O to the sphere at angle $\phi$ is $L_{eq} \cos \phi$.
   • The height of the sphere from the floor is $h(\phi) = H_{total} - L_{eq} \cos \phi = (L_{eq} + h_{floor}) - L_{eq} \cos \phi$.

3. Condition for Not Rubbing the Floor:

   • The sphere does not rub the floor if $h(\phi) \ge 0$ for all angles $\phi$ during its swing.
   • The minimum height occurs at $\phi = 0$ (the lowest point of the swing).
   • $h_{min} = h(0) = (L_{eq} + h_{floor}) - L_{eq} \cos 0 = (L_{eq} + h_{floor}) - L_{eq} = h_{floor}$.
   • Since $h_{floor} = 2$ mm, which is greater than 0, the minimum height of the sphere above the floor is $2$ mm.

4. Conclusion:

   • Because the minimum height of the sphere from the floor is always positive ($2$ mm), the sphere never rubs the floor, regardless of the initial angle $\theta$ from which it is released.

   • Therefore, any angle $\theta$ is permissible under the condition "does not rub the floor".

   • Possible Misinterpretation: If the question intended to ask for the maximum angle $\theta$ such that the sphere just touches the floor at this maximum angle, then $h(\theta_{max}) = 0$. $(L_{eq} + h_{floor}) - L_{eq} \cos \theta_{max} = 0$ $L_{eq} \cos \theta_{max} = L_{eq} + h_{floor}$ $\cos \theta_{max} = 1 + \frac{h_{floor}}{L_{eq}} = 1 + \frac{2 \times 10^{-3} \text{ m}}{4.0051 \text{ m}} \approx 1 + 0.0005$. This leads to $\cos \theta_{max} > 1$, which is physically impossible.

   • The problem statement, as given, leads to the conclusion that the sphere never rubs the floor. This suggests either the question is flawed, or it's a trick question where any angle is valid. Given the context of physics problems, a solvable scenario is usually intended. However, based strictly on the wording, the condition is always met.