How to Solve Basic Mechanics Problems
To solve a mechanics problem, you should read the problem carefully and attempt to relate it to theories you’ve learned. Draw accurate diagrams and write out the problem data. Next, apply physical principles you have learned such as Newtons Laws, and tabulate mathematical equations. Double check to make sure your equations are dimensionally homogeneous: a fancy way to say that they have similar units on either side of the equals sign. Solve your equations and check your solution if possible. Check to see whether your answer sounds reasonable in a “common sense” way.
A Step-by-Step Method
The approach above can be turned into a repeatable routine. The steps below are adapted from the problem-solving strategy for applying Newton’s laws in OpenStax University Physics.
1. List What You Know and What You Need
Write down every given quantity with its units, and state clearly what the problem asks for. This step usually tells you which physical principles apply. If forces are involved, Newton’s laws are a likely tool. If only motion is described, the kinematic equations may be enough.
2. Sketch the Situation and Draw a Free-Body Diagram
Choose the object (the “system”) you are analyzing. Draw it on its own and show every external force acting on it as an arrow: weight, normal force, tension, friction, and any applied force. Include only forces acting on the object, not forces it exerts on other things.
3. Choose Convenient Axes
Pick axes that make the math simple. It is usually best to put one axis along the direction of motion. On an incline, tilt the axes so that x runs parallel to the slope and y runs perpendicular to it.
4. Apply Newton’s Second Law in Components
Write ΣFx = max and ΣFy = may, then solve for the unknowns. Use kinematic equations if you also need distances, speeds, or times.
5. Check Units and Reasonableness
Confirm that each side of every equation has the same units. Then ask whether the answer makes physical sense, and test it against limiting cases where you already know the result.
Worked Example: A Block on a Frictionless Incline
Problem: A 2.0 kg block is released from rest on a frictionless ramp inclined at 30° to the horizontal. Find the normal force, the block’s acceleration, and its speed after it slides 2.0 m down the ramp. Use g = 9.8 m/s².
Givens: m = 2.0 kg, θ = 30°, g = 9.8 m/s², v0 = 0, distance d = 2.0 m.
Forces: Two forces act on the block: its weight w = mg = (2.0)(9.8) = 19.6 N, straight down, and the normal force N, perpendicular to the ramp. There is no friction.
Axes: Take x down the slope and y perpendicular to it. The weight splits into components:
- Parallel to the slope: wx = mg sin 30° = 19.6 × 0.500 = 9.8 N
- Perpendicular to the slope: wy = mg cos 30° = 19.6 × 0.866 ≈ 17.0 N
Newton’s second law:
- y-direction (no acceleration off the surface): N − mg cos 30° = 0, so N ≈ 17.0 N
- x-direction: mg sin 30° = ma, so a = g sin 30° = 9.8 × 0.500 = 4.9 m/s²
Kinematics: Using v² = v0² + 2ad: v² = 0 + 2(4.9)(2.0) = 19.6 m²/s², so v ≈ 4.4 m/s.
Check: Units work out to newtons and meters per second. The acceleration is less than g, as it should be on a slope. In the limiting cases θ = 0° gives a = 0, and θ = 90° gives free fall, a = g. As an independent check, sliding 2.0 m down a 30° slope drops the block 1.0 m vertically, and a frictionless drop of 1.0 m gives v = √(2gh) = √(2 × 9.8 × 1.0) ≈ 4.4 m/s, the same answer.
For more practice applying these principles, see The Resultant of Multiple Forces, and for a lab that combines momentum and energy conservation, see the Ballistic Pendulum Experiment.
Sources: OpenStax University Physics Vol. 1, 6.1: Solving Problems with Newton’s Laws; 5.6: Common Forces (weight components on an incline); 3.4: Motion with Constant Acceleration