Dynamics Methods and Non-Constant Forces

Why It Matters

JC questions often become easier when the correct method is chosen. Force equations, momentum methods, and energy methods answer different types of questions.

Definition

Non-constant force dynamics involves situations where the resultant force changes with time, position, or velocity. In such cases, constant-acceleration equations may not apply directly.

Key Representations

Momentum form of Newton’s second law:

$${\vec{F}}_{\text{net}}=\frac{d\vec{p}}{dt}$$

Impulse for varying resultant force:

$$\vec{J}=\int {\vec{F}}_{\text{net}}dt$$

Work by a variable force:

$$W=\int \vec{F}\cdot d\vec{x}$$

In one-dimensional problems, after choosing a positive direction, this becomes $W=\int F_x(x)dx$ using signed components.

Spring force magnitude in the Hooke’s law region:

$$F=kx$$

Choosing a Method

Use force equations when:

• acceleration or contact force is required,
• forces are known and can be resolved,
• the motion can be analysed instant by instant.

Use momentum methods when:

• the problem involves collisions, explosions, or recoil,
• forces are large, brief, or unknown,
• external impulse is negligible.

Use energy methods when:

• the problem asks for speed, height, work, power, or losses,
• forces vary with displacement,
• mechanical energy is conserved or losses can be accounted for.

Non-Constant Forces

Forces may vary with:

• time, such as impact forces in a collision;
• position, such as spring force;
• speed, such as drag.

Graphical methods are often useful:

• Area under a resultant force-time graph gives impulse.
• Signed area under a force-displacement graph gives work only when the graph plots the force component along the displacement.

Common Exam Points

• Do not use SUVAT when acceleration is not constant.
• Momentum conservation does not require kinetic energy conservation.
• Energy methods can include non-conservative work if losses are accounted for.
• Force-time graph area gives impulse; force-displacement graph area gives work only when the force component is plotted against displacement.

Links

• Related: dynamics
• Related: work energy and power
• Related: newtonian dynamics applications
• Related: momentum and impulse
• Related: momentum conservation and collisions
• Related: mechanical energy conservation and losses
• Related: mechanical work vector definition
• Related: work and energy transfer
• Misconception: force time graph area
• Misconception: inelastic ke conservation
• Misconception: momentum conservation system