Advertisements
Advertisements
प्रश्न
Assuming the expression for displacement of a particle starting from extreme position, explain graphically the variation of velocity and acceleration w.r.t. time.
Advertisements
उत्तर
Displacement-time graph:
i.
At extreme position, α = π/2
Displacement, x = A cos ωt
Velocity time graph:
i.
At extreme position, α = π/2
Velocity of a particle is v = - Aω sin ωt
ii. Table:
Substituting ω = 2π/T in above equation,
| Time (t) | Phase (ωt) | Velocity (v) |
| 0 | 0 | 0 |
| T/4 | π/2 | -Aω |
| T/2 | π | 0 |
| 3T/4 | 3π/2 | Aω |
| T | 2π | 0 |
Graph:
Acceleration-time graph:
i. At extreme position,α = π/2
Acceleration of a particle is,
a =-Aω2 cos ωt
ii. Table:
Substituting ω = 2π/T in above equation
| Time (t) | Phase (ωt) | Velocity (v) |
| 0 | 0 | -Aω2 |
| T/4 | π/2 | 0 |
| T/2 | π | Aω2 |
| 3T/4 | 3π/2 | 0 |
| T | 2π | -Aω2 |
iii. Graph:
Conclusions:
i. Displacement, velocity and acceleration of S.H.M. are periodic functions of time.
ii. The displacement and acceleration curves are sine curves whereas velocity curve is
cosine curve (α = 0).
iii. The phase difference between displacement and acceleration is of π radian.
iv. The phase difference between displacement and velocity and velocity and acceleration
is of π/2 radian.
v. The displacement and acceleration is maximum at extreme position whereas velocity is
minimum at the same position.
vi. All curves repeat same path after phase of 2π radian.
APPEARS IN
संबंधित प्रश्न
Hence obtain the expression for acceleration, velocity and displacement of a particle performing linear S.H.M.
A particle executing simple harmonic motion comes to rest at the extreme positions. Is the resultant force on the particle zero at these positions according to Newton's first law?
The energy of system in simple harmonic motion is given by \[E = \frac{1}{2}m \omega^2 A^2 .\] Which of the following two statements is more appropriate?
(A) The energy is increased because the amplitude is increased.
(B) The amplitude is increased because the energy is increased.
Can a pendulum clock be used in an earth-satellite?
A hollow sphere filled with water is used as the bob of a pendulum. Assume that the equation for simple pendulum is valid with the distance between the point of suspension and centre of mass of the bob acting as the effective length of the pendulum. If water slowly leaks out of the bob, how will the time period vary?
A block of known mass is suspended from a fixed support through a light spring. Can you find the time period of vertical oscillation only by measuring the extension of the spring when the block is in equilibrium?
The force acting on a particle moving along X-axis is F = −k(x − vo t) where k is a positive constant. An observer moving at a constant velocity v0 along the X-axis looks at the particle. What kind of motion does he find for the particle?
Figure represents two simple harmonic motions.
The parameter which has different values in the two motions is
A pendulum clock keeping correct time is taken to high altitudes,
A particle moves in a circular path with a continuously increasing speed. Its motion is
A particle moves on the X-axis according to the equation x = x0 sin2 ωt. The motion is simple harmonic
In a simple harmonic motion
In a simple harmonic motion
(a) the maximum potential energy equals the maximum kinetic energy
(b) the minimum potential energy equals the minimum kinetic energy
(c) the minimum potential energy equals the maximum kinetic energy
(d) the maximum potential energy equals the minimum kinetic energy
An object is released from rest. The time it takes to fall through a distance h and the speed of the object as it falls through this distance are measured with a pendulum clock. The entire apparatus is taken on the moon and the experiment is repeated
(a) the measured times are same
(b) the measured speeds are same
(c) the actual times in the fall are equal
(d) the actual speeds are equal
A particle executes simple harmonic motion with an amplitude of 10 cm and time period 6 s. At t = 0 it is at position x = 5 cm going towards positive x-direction. Write the equation for the displacement x at time t. Find the magnitude of the acceleration of the particle at t = 4 s.
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 simple pendulum of length l is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator (a) is going up with and acceleration a0(b) is going down with an acceleration a0 and (c) is moving with a uniform velocity.
A uniform rod of length l is suspended by an end and is made to undergo small oscillations. Find the length of the simple pendulum having the time period equal to that of the road.
A closed circular wire hung on a nail in a wall undergoes small oscillations of amplitude 20 and time period 2 s. Find (a) the radius of the circular wire, (b) the speed of the particle farthest away from the point of suspension as it goes through its mean position, (c) the acceleration of this particle as it goes through its mean position and (d) the acceleration of this particle when it is at an extreme position. Take g = π2 m/s2.
A particle is subjected to two simple harmonic motions of same time period in the same direction. The amplitude of the first motion is 3.0 cm and that of the second is 4.0 cm. Find the resultant amplitude if the phase difference between the motions is (a) 0°, (b) 60°, (c) 90°.
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.
If the inertial mass and gravitational mass of the simple pendulum of length l are not equal, then the time period of the simple pendulum is
Define the time period of simple harmonic motion.
What is an epoch?
Write short notes on two springs connected in series.
What is meant by simple harmonic oscillation? Give examples and explain why every simple harmonic motion is a periodic motion whereas the converse need not be true.
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")`
A simple harmonic motion is given by, x = 2.4 sin ( 4πt). If distances are expressed in cm and time in seconds, the amplitude and frequency of S.H.M. are respectively,
Motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower point is ______.
- simple harmonic motion.
- non-periodic motion.
- periodic motion.
- periodic but not S.H.M.
A weightless rigid rod with a small iron bob at the end is hinged at point A to the wall so that it can rotate in all directions. The rod is kept in the horizontal position by a vertical inextensible string of length 20 cm, fixed at its midpoint. The bob is displaced slightly, perpendicular to the plane of the rod and string. The period of small oscillations of the system in the form `(pix)/10` is ______ sec. and the value of x is ______.
(g = 10 m/s2)
