Advertisements
Advertisements
Question
The ear-ring of a lady shown in figure has a 3 cm long light suspension wire. (a) Find the time period of small oscillations if the lady is standing on the ground. (b) The lady now sits in a merry-go-round moving at 4 m/s1 in a circle of radius 2 m. Find the time period of small oscillations of the ear-ring.
Advertisements
Solution
Given,
Length of the long, light suspension wire, l = 3 cm = 0.03 m
Acceleration due to gravity, `g = 9.8 "ms"^(- 2)`
(a)Time Period \[\left( T \right)\] is given by ,
\[T = 2\pi\sqrt{\left( \frac{l}{g} \right)}\]
\[ = 2\pi\sqrt{\left( \frac{0 . 03}{9 . 8} \right)}\]
\[ = 0 . 34 \text { second}\]
(b) Velocity of merry-go-round, v = 4 `"ms"^(- 1)`
Radius of circle, r = 2 m
As the lady sits on the merry-go-round, her earring experiences centripetal acceleration.
Centripetal acceleration (a) is given by,
\[a = \frac{v^2}{r} = \frac{4^2}{2} = 8 m/ s^2\]
Resultant acceleration (A) is given by ,
\[A = \sqrt{\left( g^2 + a^2 \right)}\]
\[ = \sqrt{\left( 96 . 04 + 64 \right)}\]
\[ = 12 . 65 m/ s^2\]
Time Period,
\[T = 2\pi\sqrt{\left( \frac{l}{A} \right)}\]
\[= 2\pi\sqrt{\left( \frac{0 . 03}{12 . 65} \right)}\]
\[ = 0 . 30 \text { second }\]
APPEARS IN
RELATED QUESTIONS
The length of the second’s pendulum in a clock is increased to 4 times its initial length. Calculate the number of oscillations completed by the new pendulum in one minute.
A person goes to bed at sharp 10.00 pm every day. Is it an example of periodic motion? If yes, what is the time period? If no, why?
The total mechanical energy of a spring-mass system in simple harmonic motion is \[E = \frac{1}{2}m \omega^2 A^2 .\] Suppose the oscillating particle is replaced by another particle of double the mass while the amplitude A remains the same. The new mechanical energy will
A particle executes simple harmonic motion under the restoring force provided by a spring. The time period is T. If the spring is divided in two equal parts and one part is used to continue the simple harmonic motion, the time period will
A particle moves in a circular path with a uniform speed. Its motion is
The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitude 2 cm, 1 m s−1 and 10 m s−2 at a certain instant. Find the amplitude and the time period of the motion.
A spring stores 5 J of energy when stretched by 25 cm. It is kept vertical with the lower end fixed. A block fastened to its other end is made to undergo small oscillations. If the block makes 5 oscillations each second what is the mass of the block?
Find the time period of the motion of the particle shown in figure . Neglect the small effect of the bend near the bottom.
A uniform disc of radius r is to be suspended through a small hole made in the disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the centre for it to have minimum time period?
A body of mass 1 kg is mafe to oscillate on a spring of force constant 16 N/m. Calculate (a) Angular frequency, (b) Frequency of vibrations.
The period of oscillation of a body of mass m1 suspended from a light spring is T. When a body of mass m2 is tied to the first body and the system is made to oscillate, the period is 2T. Compare the masses m1 and m2
A 20 cm wide thin circular disc of mass 200 g is suspended to rigid support from a thin metallic string. By holding the rim of the disc, the string is twisted through 60° and released. It now performs angular oscillations of period 1 second. Calculate the maximum restoring torque generated in the string under undamped conditions. (π3 ≈ 31)
Which of the following example represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
The motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point.
The equation of motion of a particle is x = a cos (αt)2. The motion is ______.
The displacement time graph of a particle executing S.H.M. is shown in figure. Which of the following statement is/are true?
- The force is zero at `t = (T)/4`.
- The acceleration is maximum at `t = (4T)/4`.
- The velocity is maximum at `t = T/4`.
- The P.E. is equal to K.E. of oscillation at `t = T/2`.
A person normally weighing 50 kg stands on a massless platform which oscillates up and down harmonically at a frequency of 2.0 s–1 and an amplitude 5.0 cm. A weighing machine on the platform gives the persons weight against time.
- Will there be any change in weight of the body, during the oscillation?
- If answer to part (a) is yes, what will be the maximum and minimum reading in the machine and at which position?
The time period of a simple pendulum is T inside a lift when the lift is stationary. If the lift moves upwards with an acceleration `g/2`, the time period of the pendulum will be ______.
A particle performs simple harmonic motion with a period of 2 seconds. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is `1/a` s. The value of 'a' to the nearest integer is ______.
