Mathematics 9709/43 — May/June 2011

Walkthrough

The problem gives us three known quantities (force, distance, work done) and asks for the angle at which the force acts. The work done by a force depends on the component of the force along the direction of motion, which is why we multiply the force by the cosine of the angle between the force and the displacement. We start from $W = Fd\cos\theta$, substitute the given numbers, solve step by step to isolate $\cos\alpha$, and finally take the inverse cosine to find the angle. The multiplication $180 \times 50 = 9000$ shows the work that would be done if the rope were horizontal; since the actual work is less ($8200\text{ J}$), the rope must act at some angle, and the ratio gives $\cos\alpha$.

Key Takeaways

• Work done by a constant force at an angle is the component of the force in the direction of motion times the distance.
• Rearranging an equation with $\cos$ and then using the inverse cosine yields an angle.
• The cosine ratio compares the actual work to the work the full force would do along the motion.

Common Mistakes

• Using $W = Fd$ without the cosine factor.
• Mistaking $\alpha$ for the angle from the vertical rather than the angle to the horizontal (here the angle to the floor).
• Failing to use the correct calculator mode, so the angle comes out in the wrong unit.

Things to Be Careful About

• The calculator must be in degree mode, since the answer is required in degrees.
• Keep the units consistent (N, m, J) - since the work is given in joules and distances in metres, no conversion is needed.
• The angle to the floor is the angle between the rope and the horizontal, exactly the angle used in $\cos\alpha$. Only the horizontal component of the tension does work.
• Rounding: give the answer to one decimal place as shown ($24.3^\circ$), matching the mark scheme.