The Frequency of Vibration of a String Depends on the Length L Between the Nodes, the Tension F in the String and Its Mass per Unit Length M. Guess the Expression for - Physics

Advertisements
Advertisements
Sum

The frequency of vibration of a string depends on the 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.

Advertisements

Solution

Frequency, f \[\propto\] LaFbmc
f = kLaFbmc   ...(1)
Dimension of [f] = [T−1]

Dimension of the right side components:
[L] = [L]
[F] = [MLT−2]
[m] = [ML−1]

Writing equation (1) in dimensional form, we get:
[T−1] = [L]a [MLT−2]b [ML−1]c
[M0L0T−1] = [Mb + c La + b − c T−2b]

Equating the dimensions of both sides, we get:
b + c = 0               ....(i)
a + b −c = 0      ....(ii)
−2b = −1              ....(iii) 
Solving equations (i), (ii) and (iii), we get:

\[a = - 1, b = \frac{1}{2} \text{ and } c = \left( - \frac{1}{2} \right)\] 
∴ Frequency, f = kL−1 F1/2m−1/2

\[= \frac{k}{L} F^{1/2} m^\left( - \frac{1}{2} \right) = \frac{k}{L} \sqrt{\left( \frac{F}{m} \right)}\]
\[f = \frac{k}{L} \sqrt{\left( \frac{F}{m} \right)}\]

  Is there an error in this question or solution?
Chapter 1: Introduction to Physics - Exercise [Page 10]

APPEARS IN

HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 1 Introduction to Physics
Exercise | Q 17 | Page 10

RELATED QUESTIONS

The normal duration of I.Sc. Physics practical period in Indian colleges is 100 minutes. Express this period in microcenturies. 1 microcentury = 106 × 100 years. How many microcenturies did you sleep yesterday?


Can we have physical quantities having magnitude and direction which are not vectors?


The length and the radius of a cylinder measured with a slide callipers are found to be 4.54 cm and 1.75 cm respectively. Calculate the volume of the cylinder.


The length of the string of a simple pendulum is measured with a metre scale to be 90.0 cm. The radius of the bod plus the length of the hook is calculated to be 2.13 cm using measurements with a slide callipers. What is the effective length of the pendulum? (The effective length is defined as the distance between the point of suspension and the centre of the bob.)


Laws of conservation of linear momentum states that linear momentum of a system


Consider a gravity-free hall in which an experimenter of mass 50 kg is resting on a 5 kg pillow, 8 ft above the floor of the hall. He pushes the pillow down so that it starts falling at a speed of 8 ft/s. The pillow makes a perfectly elastic collision with the floor, rebounds and reaches the experimenter's head. The time elapsed in the process is ______ s.


A ball of mass m = 20 kg released from height h = 10m falls on the Earth's surface. The speed of the Earth when the ball reaches the Earth's surface is ______ × 10-23 m/s.


A body starts moving from its origin at t = 0 with a velocity of `5hati` in the x-y plane under the action of force producing an acceleration of `(3hati + 2hatj)` m/s2, then y-co-ordinate is ______ m of the body when x-co-ordinate is 84 m.


A wedge of mass M = 2m0 rests on a smooth horizontal plane. A small block of mass m0 rests over it at left end A as shown in the figure. A sharp impulse is applied on the block, due to which it starts moving to the right with velocity v0 = 6 m/s. At the highest point of its trajectory, the block collides with a particle of the same mass m0 moving vertically downwards with velocity v = 2 m/s, and gets stuck with it. If the combined mass lands at the end point A of the body of mass M, the length l is ______ cm. Neglect friction, take g = 10 m/s2.

   


Share
Notifications



      Forgot password?
Use app×