Question

a small mass is suspended from a long thread to form a pendulum. The period (T) of the oscillations depend on the mass (M) the length (L) od the thread and acceleration (g) of free fall at the place concern T=kmx ly g2 where x,y and z are unknown and k = 2piers.
1, find the value of x,y,z.
2, hence derive an expression relating t, m, l and g
Consider the formula b=ut at, find the dimension of the equation.

Answer 1

Explanations & Calculations

a)

$\qquad\qquad \begin{aligned} \small T&=\small 2\pi.m^xl^yg^z\\ \small [T]&=\small T\to M^0L^0T^1\\ \small [m]&=\small M,\,\,[l]=L,\,\,[g]= LT^{-2}\\ \\ \small M^0L^0T^1&:\small M^xL^y(LT^{-2})^z\\ &: \small M^x L^{(y+z)}T^{-2z}\\ \\ \small M: \,\:0&= x\\ \small L:\,\; 0&=\small y+z\to y = -z\\ \small T:\,\, 1 &=\small -2z\\ \\ \therefore\,, \;\small x=0,\, y &=\frac{1}{2},\,z=\frac{-1}{2} \\ \therefore, \small T&=\small 2\pi l^{\frac{1}{2}}g^{\frac{-1}{2}}\\ &=\small 2\pi \sqrt{\frac{l}{g}} \end{aligned}$

b)

$\qquad\qquad \begin{aligned} \small b&=\small ut\\ \small [b]&=\small L\\ \small [ut]&=\small LT^{-1}\times T=L \end{aligned}$

• Therefore, the dimension of the equation is the dimension of the length: L.