## Question

A mass *m* is suspended at the end of a massless wire of length *L* and cross-sectional area *A*. If *Y* is the Young’s modulus of the material of the wire, the frequency of oscillations along the vertical line is given by

### Solution

Let *l* be the increase in the length of the wire due to the force *F* = *mg* (see fig.) Then

By definition, Young’s modulus is

This is the force acting upwards in the

equilibrium state. If the mass is pulled

down a little through a distance *x*, so

that the total extension in the string

is (*l* + *x*), then the force in the wire acting upwards will be

And downward force is *F* = *mg*. The restoring force is the net downward force. Hence,

is the angular frequency of the resulting motion which is simple harmonic. Thus,

#### SIMILAR QUESTIONS

Figure (a) shows a spring of force constant *k* fixed at one end and carrying a mass *m* at the other end placed on a horizontal frictionless surface. The spring is stretched by a force *F*. figure (b) shows the same spring with both ends free and a mass *m* fixed at each free end. Each of the spring is stretched by the same force *F*. the mass is case (a) and the masses in case (b) are then released.

Which of the following statements is/are true?

A simple pendulum of bob mass *m* is oscillating with an angular amplitudeα* _{m}* (in radius). The maximum tension in the string is

A particle is executing simple harmonic motion. Its displacement is given by

where *x* is in cm and *t* in seconds. How long will the particle take to move from the position of equilibrium to the position of maximum displacement?

A simple pendulum is moving simple harmonically with a period of 6 s between two extreme position B and C about a point O. if the angular distance between B and C is 10 cm, how long will the pendulum take to move from position C to a position D exactly midway between O and C.

A horizontal platform with an object placed on it is executing SHM in the vertical direction. The amplitude of oscillation is 2.5 cm. What must be the least period of these oscillations so that the object is not detached from the platform? Take *g* = 10 ms^{ –2}. ^{ }

A spring with no mass attached to it hangs from a rigid support. A mass *m*is now hung on the lower end to the spring. The mass is supported on a platform so that the spring remains relaxed. The supporting platform is then suddenly removed and the mass begins to oscillate. The lowest position of the mass during the oscillation is 5 cm below the place where it was resting on the platform. What is the angular frequency of oscillation? Take g = 10 ms^{–2}.

When a mass *m* is hung from the lower of a spring of negligible mass, an extension *x* is produced in the spring. The mass is set into vertical oscillations. The time period of oscillation is

A small spherical steel ball is placed a little away from the centre of a large concave mirror of radius of curvature *R* = 2.5 *m. *The ball is then released. What is the time period of the motion? Neglect friction and take

Two masses *m*_{1} and *m*_{2} are suspended together by a massless spring of force constant *k* (see fig). When the masses are in equilibrium, mass *m*_{1} is removed without disturbing the system. The angular frequency of oscillation of mass *m*_{2} is

A test tube of cross-sectional area *a* has some lead shots in it. The total mass is *m*. It floats upright in a liquid of density *d*. When pushed down a little and released, it oscillates up and down with a period *T*. Use dimensional considerations and choose the correct relationship from the following.