To model the time taken for a small sphere released from rest to roll down a smooth…

PHYSICS · Subjective · A Level · Physics · Measurements and their errors · Measurements and their errors

Problem

To model the time

t

taken for a small sphere released from rest to roll down a smooth straight chute of length

l

inclined at an angle

\theta

to the horizontal, the empirical relation

t = k \sqrt{\dfrac{l}{g\sin\theta}}

is proposed, where

g

is the acceleration due to gravity and

k

is an unknown constant. (a) Using dimensional analysis, determine the dimensions of

k

. (b) Express these dimensions in (i) the SI system and (ii) the CGS system of units. (c) In an experiment with

l = 2.0\,\text{m}

and

\theta = 30^{\circ}

the measured time is

1.55\,\text{s}

. Taking

g = 9.8\,\text{m\,s}^{-2}

, calculate the numerical value of

k

for this set-up and comment on whether

k

can reasonably be taken as a universal constant for such motions.