JEE Physics Units Kinematics and Gravitation Notes Set 01

Units and Dimensions

Dimensions show how a quantity depends on mass M, length L and time T. A physical equation must have the same dimensions on both sides, and only quantities with the same dimensions can be added or subtracted. The argument of a sine, a logarithm or an exponential must be dimensionless.

To derive a formula by dimensions, write the quantity as a product of powers of the variables, equate the powers of M, L and T and solve the equations. The constant of proportionality cannot be found this way.

Errors

1. Sum or difference: absolute errors add, \(\Delta Z = \Delta A + \Delta B\).
2. Product or quotient: fractional errors add, \(\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}\).
3. Power: for \(Z = A^n\), \(\dfrac{\Delta Z}{Z} = n\dfrac{\Delta A}{A}\).
4. Least count of a vernier = one main scale division minus one vernier division.

Worked example. For \(\rho = \dfrac{M}{\pi r^2 \ell}\) with 1% error in each of M, r and \(\ell\), the maximum error is \(1 + 2 + 1 = 4\%\).

Kinematics

Distance is the length of the path travelled and displacement is the straight-line change of position. In a river-crossing problem the crossing time depends only on the velocity component across the river.

Worked example. A ball thrown upward with speed u covers distances \(u - g(n - \tfrac12)\) in the n-th second, as long as it is still rising.

Gravitation

Gravitational force is an action-reaction pair, so the two bodies exert equal and opposite forces on each other. For a body lifted to a height comparable with R, use \(\Delta U = GMm\left(\dfrac{1}{R} - \dfrac{1}{R+h}\right)\), not mgh. For particles moving in a circle under their mutual attraction, equate the net inward force to \(\dfrac{mv^2}{r}\).