Physics 5054/12 — October/November 2020

Walkthrough

1. State the equilibrium condition: The nail is fixed and does not move, meaning it is in static equilibrium. By Newton's first law, the resultant force acting on it is zero.
2. Determine the direction of each force:
   • Weight $W$: Always acts vertically downwards towards the centre of the Earth.
   • Tension $T$: Acts along the string, pulling the nail away from it. The string goes upwards and to the right at $40^\circ$ to the vertical, so $T$ must point upwards and to the right.
   • Reaction force $R$: Must balance the other two forces. Since $T$ has a horizontal component to the right, $R$ must have a horizontal component to the left to ensure the net horizontal force is zero.
3. Apply the triangle of forces: When three forces are in equilibrium, they can be drawn as a closed triangle if placed head-to-tail. The vector sum $\vec{W} + \vec{T} + \vec{R} = 0$ means that following the arrows in sequence must return you to the starting point.
4. Evaluate the options:
   • A: $W$ is down, $R$ is left, $T$ is up-right. Tracing head-to-tail (down $\rightarrow$ left $\rightarrow$ up-right) closes the triangle perfectly. All directions match our analysis.
   • B: $T$ is drawn pointing down-left, which contradicts the string pulling the nail up and right.
   • C: $R$ is drawn pointing down-right. This would add to the rightward pull of $T$ instead of balancing it, so the horizontal resultant cannot be zero.
   • D: $T$ is drawn pointing down-left, again contradicting the physical setup.
5. Conclusion: Diagram A is the only correct representation.

Key Takeaways

• An object in equilibrium has a resultant force of zero.
• Three concurrent forces in equilibrium can be represented as a closed triangle of forces when drawn head-to-tail.
• Tension in a string always acts along the string, pulling away from the object it is attached to.
• The reaction force from a wall must have a component normal to the surface to balance any lateral pull.

Common Mistakes

• Drawing tension in the wrong direction: Tension pulls away from the object along the string. Candidates sometimes draw it pointing down the string towards the anchor point.
• Forgetting the horizontal balance: Some candidates only check vertical components and miss that $R$ must point leftwards to balance the rightward horizontal component of $T$.
• Misinterpreting the triangle: Forgetting that the arrows must be placed head-to-tail to form a closed loop; checking only the directions without verifying the head-to-tail closure leads to selecting diagrams like C where the vectors do not form a closed polygon.

Things to Be Careful About

• Vector direction vs. line of action: The line of action of $T$ is along the string, but the arrow must point away from the nail (up and right). The angle $40^\circ$ to the vertical confirms it is not purely horizontal or vertical.
• Closed triangle rule: The triangle of forces requires head-to-tail placement. Just having the correct directions is not enough; the vectors must connect tip-to-tail to sum to zero.
• Wall reaction direction: The force $R$ from the wall is not necessarily purely horizontal. Because the nail is "fixed firmly", the wall can exert both a normal reaction and a frictional force, so $R$ can point at any angle necessary to maintain equilibrium (here, left and slightly down or up depending on magnitudes, but definitely leftwards).