4 \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_512_983_267_580} A uniform sphere rests on a horizontal plane. The sphere has centre \(O\), radius 0.6 m and weight 36 N . A uniform rod \(A B\), of weight 14 N and length 1 m , rests with \(A\) in contact with the plane and \(B\) in contact with the sphere at the end of a horizontal diameter. The point of contact of the sphere with the plane is \(C\), and \(A , B , C\) and \(O\) lie in the same vertical plane (see diagram). The contacts at \(A , B\) and \(C\) are rough and the system is in equilibrium. By taking moments about \(C\) for the system, show that the magnitude of the normal contact force at \(A\) is 10 N . Show that the magnitudes of the frictional forces at \(A , B\) and \(C\) are equal. The coefficients of friction at \(A , B\) and \(C\) are all equal to \(\mu\). Find the smallest possible value of \(\mu\).

## Question 4:

| Working/Answer | Mark | Guidance |
|---|---|---|
| $1.4R_A = 1.0\times 14$, $R_A = 10$ **A.G.** | M1 A1 | Find $R_A$ by taking moments about $C$ for system. Part mark: 2 |
| $F_B = F_C$ **A.G.** | B1 | Deduce by taking moments about $O$ for sphere |
| $F_A = F_C$ **A.G.** | B1 | Resolve horizontally for system. Part mark: 2 |
| $F_B = 14 - R_A = 4$ | M1 A1 | Find any $F$ by e.g. vertical resolution for $AB$ |
| *or* $F_A = (0.8R_A - 0.4\times14)/0.6 = 4$ | (M1 A1) | *or* taking moments about $B$ for $AB$ |
| $R_B = F_A$ or $F_C\ [= 4]$ | M1 | Find $R_B$ by e.g. horizontal resolution for rod or sphere |
| $R_C = 36 + F_B$ or $50 - R_A = 40$ | M1 A1 | Find $R_C$ by e.g. vertical resolution for sphere or system |
| $\mu_{min} = \max\{F_A/R_A,\ F_B/R_B,\ F_C/R_C\}$ | M1 | |
| $= \max\{4/10,\ 4/4,\ 4/40\} = 1$ | A1 | Part mark: 7. Total: **[11]** |

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\includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_512_983_267_580}

A uniform sphere rests on a horizontal plane. The sphere has centre $O$, radius 0.6 m and weight 36 N . A uniform rod $A B$, of weight 14 N and length 1 m , rests with $A$ in contact with the plane and $B$ in contact with the sphere at the end of a horizontal diameter. The point of contact of the sphere with the plane is $C$, and $A , B , C$ and $O$ lie in the same vertical plane (see diagram). The contacts at $A , B$ and $C$ are rough and the system is in equilibrium. By taking moments about $C$ for the system, show that the magnitude of the normal contact force at $A$ is 10 N .

Show that the magnitudes of the frictional forces at $A , B$ and $C$ are equal.

The coefficients of friction at $A , B$ and $C$ are all equal to $\mu$. Find the smallest possible value of $\mu$.

\hfill \mbox{\textit{CAIE FP2 2009 Q4 [11]}}