Mathematics 9709/41 — October/November 2022

Walkthrough

The problem states that four coplanar forces act at a point and are in equilibrium. Equilibrium means the net force is zero, so we can split this into two independent conditions: the sum of horizontal components is zero, and the sum of vertical components is zero.

Step 1: Identify the direction and components of each force.

From the diagram:

• $Q\text{ N}$ acts vertically upwards along the positive $y$-axis. Its horizontal component is $0$ and its vertical component is $+Q$.
• $22\text{ N}$ acts horizontally to the right along the positive $x$-axis. Its horizontal component is $+22$ and its vertical component is $0$.
• $16\text{ N}$ acts in the first quadrant at $55°$ above the positive $x$-axis. Its horizontal component is $+16\cos 55°$ and its vertical component is $+16\sin 55°$.
• $P\text{ N}$ acts in the third quadrant at $25°$ below the negative $x$-axis. This means it points left and down. Its horizontal component is $-P\cos 25°$ (negative because it points left) and its vertical component is $-P\sin 25°$ (negative because it points down).

Step 2: Resolve horizontally.

Sum of horizontal components $= 0$:

$$22 + 16\cos 55° - P\cos 25° = 0$$

Rearranging gives:

$$P\cos 25° = 22 + 16\cos 55°$$

This is one equation with one unknown ($P$), so we can solve for $P$ directly.

Step 3: Resolve vertically.

Sum of vertical components $= 0$:

$$Q + 16\sin 55° - P\sin 25° = 0$$

Rearranging gives:

$$Q + 16\sin 55° = P\sin 25°$$

Now that we know $P$, we can substitute to find $Q$.

Step 4: Calculate $P$.

$$P = \frac{22 + 16\cos 55°}{\cos 25°} = \frac{22 + 9.1772}{0.9063} = \frac{31.1772}{0.9063} = 34.4003$$

So $P \approx 34.4\text{ N}$.

Step 5: Calculate $Q$.

$$Q = P\sin 25° - 16\sin 55° = 34.4003 \times 0.4226 - 16 \times 0.8192 = 14.5381 - 13.1064 = 1.4317$$

So $Q \approx 1.43\text{ N}$.

Key Takeaways

• When a system of coplanar forces is in equilibrium, both the horizontal and vertical components must independently sum to zero.
• Resolving forces into perpendicular components is the standard method for solving equilibrium problems with forces at angles.
• Always carefully determine the sign (positive or negative) of each component based on the direction the force points relative to the chosen positive axes.
• When a force makes an angle $\theta$ with a horizontal or vertical reference line, use $\cos\theta$ for the component along that reference line and $\sin\theta$ for the perpendicular component.

Common Mistakes

• Sign errors in components: Forgetting that $P$ acts in the third quadrant means both its horizontal and vertical components are negative. A common error is writing $+P\cos 25°$ or $+P\sin 25°$ instead of $-P\cos 25°$ and $-P\sin 25°$.
• Mixing up sin and cos: Using $\sin$ where $\cos$ should be used (or vice versa) when resolving forces at an angle. Remember: the component along the reference line (the one the angle is measured from) uses $\cos$, and the perpendicular component uses $\sin$.
• Forgetting a term: The mark scheme requires 3 terms in the resolution equations. Students sometimes omit one of the forces, especially $Q$ in the horizontal equation (which is zero) or $22$ in the vertical equation (which is also zero).
• Rounding too early: Carrying only 2 or 3 decimal places in intermediate steps can lead to an incorrect final answer for $Q$. Keep at least 4-5 significant figures during calculation.

Things to Be Careful About

• Direction of $P$: The diagram shows $P$ making $25°$ with the negative $x$-axis (dashed line) in the third quadrant. This means $P$ points left and down, so both components are negative. Do not confuse this with an angle measured from the positive $x$-axis.
• Sign conventions: Choose a consistent sign convention (e.g., right and up positive) and apply it uniformly to all forces. The mark scheme allows sign errors in the initial resolution attempt (M1), but the final equations must be correct (A1).
• Precision of final answers: The mark scheme gives $P = 34.4$ and $Q = 1.43$ to 3 significant figures. Using more precise intermediate values ($P = 34.40025941$, $Q = 1.431745128$) ensures the final answers are correct to the required precision.
• Equilibrium condition: Remember that equilibrium requires the vector sum to be zero, not just the magnitudes. This is why we must resolve into components rather than simply adding magnitudes.