Subjective Type

The frequency of vibration of a string depends on teh length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.

Solution

Frequency $$f=KL^aF^bM^cM=$$Mass/unit length, $$L=$$length, $$F=$$tension(forcE)
Dimension of $$f=[T^{-1}]$$
Dimension of right side,
$$L^a=[L^a], F^b=[MLT^{-2}]^b, M^c=[ML^{-1}]^c$$
$$\therefore [T^{-1}]=K[L]^3[MLT^{-2}]^b, M^c=[ML^{-1}]^c$$
$$M^0L^0T^{-1}=KM^{b+c}L^{a+b+c}T^{-2b}$$
Equating the dimensions of both sides,
$$\therefore b+c=0$$ .......$$(1)$$
$$-c+a+b=0$$ .......$$(2)$$
$$-2b=-1$$ ........$$(3)$$
Solving the equations we get,
$$a=-1, b=1/2$$ and $$c=-1/2$$
$$\therefore$$ So, frequency $$f=KL^{-1}F^{1/2}M^{-1/2}=\dfrac{K}{L}F^{1/2}M^{-1/2}=\dfrac{K}{L}=\sqrt{\dfrac{F}{M}}$$.

SIMILAR QUESTIONS

Units and Dimensions

Find the dimension of linear momentum.

Units and Dimensions

Find the dimensions of frequency.

Units and Dimensions

Find the dimensions of pressure.

Units and Dimensions

Find the dimensions of electrifc field E. The relevant equations are $$F=qE, F=qvB$$ and $$B=\dfrac{\mu_oI}{2\pi a}$$ where F is force, q is charge, v is speed, I is current, and a is diatance.

Units and Dimensions

Find the dimensions of magnefic field B. The relevant equations are $$F=qE, F=qvB$$ and $$B=\dfrac{\mu_oI}{2\pi a}$$; where F is force, q is charge, v is speed, I is current, and a is distance.

Units and Dimensions

Find the dimensions of magnetic permeability $$\mu_o$$. The relevant equations are $$F=qE, F=qvB$$ and $$B=\dfrac{\mu_oI}{2\pi a}$$; where F is force, q is charge, v is speed, I is current, and a is distance.

Units and Dimensions

Find the dimensions of electric dipole moment p. The defining equations are $$p=q.d$$ and $$M=IA$$; where d is distance, A is area, q is charge and I is current.

Units and Dimensions

Find the dimensions of magnetic dipole moment M. The defining equations are $$p=q.d$$ and $$M=IA$$; where d is distance, A is area, q is charge and I is current.

Units and Dimensions

Which of the following pairs has quantities of the same dimensions ?

Units and Dimensions

If $$v=$$velocity of a body $$c=$$ speed of light Then the dimensions of $$\dfrac{v}{c}$$ is:

Contact Details