A cylindrical tube rolling down a slope of inclination θ moves a distance L in time T. The equation relating these quantities is

Question 13

A cylindrical tube rolling down a slope of inclination θ moves a distance L in time T. The equation relating these quantities is

L (3 + a²/P) = QT² sinθ

where a is the internal radius of the tube and P and Q are constants.

Which line gives the correct units for P and Q?

P                      Q

A          m²                          m²s^-2

B          m²                          m s^-2

C         m²                          m³s^-2

D         m³                          m s^-2

Reference: Past Exam Paper – November 2011 Paper 11 Q4 & Paper 13 Q5

Solution:

Answer: B.

For equation to be homogeneous (and hence correct), the overall units of each term should be the same on both sides of the equation.

Expanding the left hand side gives 3L + a²L/P

The unit of 3L should be the same as the overall unit of a²L/P.

Unit of L: m    (since it is a distance)

So all the different terms in the equation should have the same overall unit of m.

Considering the term a²L/P:

[a²L/P] = [m²][m] / [P] = m                 (a is the internal radius and so, has unit m)

[P] = [m³] / [m] = m²

Now consider the term QT²sinθ on the right hand side. The overall unit should also be m.

sin θ has no units.

[T²]: s² since T is the time

[Q][s²] = m

[Q] = m s^-2