Advertisements
Advertisements
Question
Consider the Earth as a homogeneous sphere of radius R and a straight hole is bored in it through its centre. Show that a particle dropped into the hole will execute a simple harmonic motion such that its time period is
T = `2π sqrt("R"/"g")`
Advertisements
Solution
Oscillations of a particle dropped in a tunnel along the diameter of the earth.
Consider earth to be a sphere of radius R and centre O. A straight tunnel is dug along the diameter of the earth. Let ‘g’ be the value of acceleration due to gravity at the surface of the earth.
Suppose a body of mass ‘m’ is dropped into the tunnel and it is at point R i.e., at a depth d below the surface of the earth at any instant.
If g’ is the acceleration due to gravity at P.
then R − d = y
∴ g’ = `"g"("y"/"R")`
Force acting on the body a point P is
F = − mg’ = `-"mg"/"R""y"` i.e., `"F" ∝ "y"`
Negative sign indicates that the force acts in the opposite direction of displacement.
Thus the body will execute SHM with force constant, k = `"mg"/"R"`
The period of oscillation of the body will be T = `2π sqrt("m"/"k") = 2π sqrt ("m"/("mg"//"R"))`
T = `2π sqrt("R"/"g")`
APPEARS IN
RELATED QUESTIONS
A body of mass 1 kg is made to oscillate on a spring of force constant 16 N/m. Calculate:
a) Angular frequency
b) frequency of vibration.
Can simple harmonic motion take place in a non-inertial frame? If yes, should the ratio of the force applied with the displacement be constant?
A platoon of soldiers marches on a road in steps according to the sound of a marching band. The band is stopped and the soldiers are ordered to break the steps while crossing a bridge. Why?
The pendulum of a certain clock has time period 2.04 s. How fast or slow does the clock run during 24 hours?
A pendulum clock giving correct time at a place where g = 9.800 m/s2 is taken to another place where it loses 24 seconds during 24 hours. Find the value of g at this new place.
A uniform disc of mass m and radius r is suspended through a wire attached to its centre. If the time period of the torsional oscillations be T, what is the torsional constant of the wire?
A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of 45° with the X-axis. The two motions are given by x = x0 sin ωt and s = s0 sin ωt. Find the amplitude of the resultant motion.
What is an epoch?
A body oscillates with SHM according to the equation x = 5 cos `(2π"t" + π/4)`. Its instantaneous displacement at t = 1 sec is:
If x = `5 sin (pi t + pi/3) m` represents the motion of a particle executing simple harmonic motion, the amplitude and time period of motion, respectively, are ______.
