Mechanics Practice Problems

Work the problem before you touch a number

List every quantity with its symbol and its unit, then convert to base SI before substituting. Mechanics formulas want kilograms, metres, seconds and newtons, while the questions quote whatever unit the matching calculator displays: extension in centimetres, area in square centimetres, density in grams per cubic centimetre.

Then pick the unknown and rearrange symbolically, before any digits go in. F = ma turned into a = F/m is one step you can check by eye. The same rearrangement done halfway through a substitution is where a division quietly becomes a multiplication.

Where mechanics answers go wrong

• Mass left in grams. The kinetic energy of 250 g at 10 m/s is 0.5 × 0.25 × 100 = 12.5 J. Leaving the mass as 250 returns 12,500 J, a clean factor of a thousand that still reads as a plausible energy.
• Area left in square centimetres. 25 cm² is 0.0025 m², not 25. A 500 N force spread over it is 200 kPa, and keeping the area as 25 gives 20 Pa, out by 10⁴.
• Extension left in centimetres. A 500 N/m spring pulled 5 cm needs 500 × 0.05 = 25 N. The centimetre figure gives 2500 N.
• Sine where cosine belongs. Torque takes the sine of the angle between the lever arm and the force, work takes the cosine of the angle between force and displacement. At 30 degrees the two differ by a factor of 1.73, and at 90 degrees one is a maximum and the other is zero.
• Density in the wrong system. Aluminium is 2.7 g/cm³ and 2700 kg/m³. Both are right, and they are not interchangeable.

Check the answer without redoing it

Anchor the order of magnitude first. A 70 kg person weighs about 690 N. A 1 kg mass dropped 1 metre arrives carrying about 9.8 J, at 4.4 m/s. A 1500 kg car at 30 m/s holds 675 kJ. An answer sitting a factor of ten from the nearest anchor is a conversion error far more often than a physics error.

Then use the exponent each quantity carries. Kinetic energy and centripetal force both go as speed squared, so a 10 percent slip in speed becomes 21 percent in the result, while momentum is linear and stays at 10 percent. Spring force is linear in extension and stored energy goes as extension squared, so an answer that moves by the wrong factor when an input doubles has lost a square.

Newton metres are not joules

Torque and energy share dimensions and separate units by convention: torque is quoted in N·m and energy in joules, because a newton metre of torque is a force acting at right angles to a lever arm rather than a force acting through a distance. Momentum has no named unit at all, and kg·m/s and N·s are the same thing written two ways.

Standard gravity is 9.80665 m/s² by definition, usually shortened to 9.81, while the true local value runs from about 9.78 m/s² at the equator to 9.83 m/s² at the poles. Rounding it to 9.8 shifts an answer by roughly 0.1 percent, inside the tolerance here and outside what four significant figures would claim.