Problems with Vectors at an Angle
Why angles matter in physics
Many physical quantities depend not only on the size of a force, but also on its direction. A force can be resolved into components, produce a turning effect, or contribute to motion depending on the angle at which it acts.
Rather than treating these relationships as isolated formulas, the app lets students adjust sliders and immediately see the corresponding vectors, components and numerical results change.
1. Resolving weight on a slope
The first activity shows a box resting on a frictionless slope. The weight of the box acts vertically downward, but it can be resolved into two useful components:
- A component parallel to the slope: $mg \sin\theta$
- A component perpendicular to the slope: $mg \cos\theta$
The parallel component causes the box to accelerate down the slope. Since friction is ignored, Newton's second law gives:
$$F_{\text{net}} = mg \sin\theta \\ a = g \sin\theta$$Students can change the slope angle and mass while observing the force vectors and acceleration update in real time. This makes it clear that increasing the slope angle increases the component of weight acting down the slope.
2. Investigating the turning effect of a force
The second activity uses a wrench and nut to demonstrate moments, also known as torque. The turning effect depends on the applied force $F$, the distance from the pivot $r$ and the angle between the force and the wrench $\theta$.
$$\tau = Fr \sin\theta$$The app separates the applied force into components parallel and perpendicular to the wrench. Only the perpendicular component produces a turning effect. This explains why:
- A force at 90 degrees produces the greatest moment.
- A longer wrench produces a greater moment.
- A force along the handle produces no turning effect.
By adjusting the angle, force and wrench length, students can see the relationship between leverage and rotational motion.
3. Understanding work done
The final activity shows how a force $F$ does work $W$ when it causes an object to move through a displacement $d$. Only the component of the force in the direction of motion contributes to the work done.
$$W = Fd \cos\theta$$The displacement vector changes length as the displacement slider is adjusted. Students can explore several important cases:
- At 0 degrees, the force does maximum positive work.
- At 90 degrees, the force does zero work.
- Above 90 degrees, the force does negative work.
This visual approach helps students understand why a large force does not always mean a large amount of work.
Designed for active learning
The app is designed to support experimentation. Students can adjust sliders, compare different situations and observe how the diagrams respond. The vector diagrams use colour and dashed component lines to distinguish the original force from its resolved components.
Each activity also includes optional worked examples and practice questions. Worked solutions update with the current slider values, while practice questions allow students to test their understanding independently.