Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
Advertisement Remove all ads

A Block Suspended from a Vertical Spring is in Equilibrium. Show that the Extension of the Spring Equals the Length of an Equivalent Simple Pendulum - Physics

Sum

A block suspended from a vertical spring is in equilibrium. Show that the extension of the spring equals the length of an equivalent simple pendulum, i.e., a pendulum having frequency same as that of the block.

Advertisement Remove all ads

Solution

An equivalent simple pendulum has same time period as that of the spring mass system.
The time period of a simple pendulum is given by,

\[T_p = 2\pi\sqrt{\left( \frac{l}{g} \right)}\]

where l is the length of the pendulum, and
           g is acceleration due to gravity.

Time period of the spring is given by,

\[T_s = 2\pi\sqrt{\left( \frac{m}{k} \right)}\]

where is the mass, and 
           is the spring constant.

Let x be the extension of the spring.
For small frequency, TP ​can be taken as equal to TS.    

\[\Rightarrow \sqrt{\left( \frac{l}{g} \right)} = \sqrt{\left( \frac{m}{k} \right)}\]

\[ \Rightarrow \left( \frac{l}{g} \right) = \left( \frac{m}{k} \right)\]

\[ \Rightarrow l = \frac{mg}{k} = \frac{F}{k} = x\]

(\[\because\] restoring force = weight = mg

\[\therefore\] l = x (proved)

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

HC Verma Class 11, 12 Concepts of Physics 1
Chapter 12 Simple Harmonics Motion
Q 10 | Page 252
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×