Advertisements
Advertisements
प्रश्न
A small block oscillates back and forth on a smooth concave surface of radius R in Figure. Find the time period of small oscillation.
Advertisements
उत्तर
It is given that R is the radius of the concave surface.
Let N be the normal reaction force.
Driving force, F = mg sin θ
Comparing the expression for driving force with the expression, F = ma, we get:
Acceleration, a = g sin θ
Since the value of θ is very small,
∴ sin θ → θ
∴ Acceleration, a = gθ
Let x be the displacement of the body from mean position.
\[\therefore \theta = \frac{x}{R}\]
\[ \Rightarrow a = g\theta = g\left( \frac{x}{R} \right)\]
\[ \Rightarrow \left( \frac{a}{x} \right) = \left( \frac{g}{R} \right)\]
\[\Rightarrow a = x\frac{g}{R}\]
As acceleration is directly proportional to the displacement. Hence, the body will execute S.H.M.
Time period \[\left( T \right)\] is given by,
\[T = 2\pi\sqrt{\frac{\text { displacement }}{\text { Acceleration }}}\]
\[= 2\pi\sqrt{\frac{x}{gx/R}} = 2\pi\sqrt{\frac{R}{g}}\]
APPEARS IN
संबंधित प्रश्न
A particle executes simple harmonic motion. If you are told that its velocity at this instant is zero, can you say what is its displacement? If you are told that its velocity at this instant is maximum, can you say what is its displacement?
Can the potential energy in a simple harmonic motion be negative? Will it be so if we choose zero potential energy at some point other than the mean position?
A pendulum clock gives correct time at the equator. Will it gain time or loose time as it is taken to the poles?
The displacement of a particle in simple harmonic motion in one time period is
A particle moves on the X-axis according to the equation x = A + B sin ωt. The motion is simple harmonic with amplitude
A wall clock uses a vertical spring-mass system to measure the time. Each time the mass reaches an extreme position, the clock advances by a second. The clock gives correct time at the equator. If the clock is taken to the poles it will
A pendulum clock keeping correct time is taken to high altitudes,
Which of the following quantities are always zero in a simple harmonic motion?
(a) \[\vec{F} \times \vec{a} .\]
(b) \[\vec{v} \times \vec{r} .\]
(c) \[\vec{a} \times \vec{r} .\]
(d) \[\vec{F} \times \vec{r} .\]
For a particle executing simple harmonic motion, the acceleration is proportional to
In a simple harmonic motion
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 simple pendulum of length 40 cm is taken inside a deep mine. Assume for the time being that the mine is 1600 km deep. Calculate the time period of the pendulum there. Radius of the earth = 6400 km.
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.
The length of a second’s pendulum on the surface of the Earth is 0.9 m. The length of the same pendulum on the surface of planet X such that the acceleration of the planet X is n times greater than the Earth is
Define the time period of simple harmonic motion.
Write short notes on two springs connected in series.
Consider two simple harmonic motion along the x and y-axis having the same frequencies but different amplitudes as x = A sin (ωt + φ) (along x-axis) and y = B sin ωt (along y-axis). Then show that
`"x"^2/"A"^2 + "y"^2/"B"^2 - (2"xy")/"AB" cos φ = sin^2 φ`
and also discuss the special cases when
- φ = 0
- φ = π
- φ = `π/2`
- φ = `π/2` and A = B
- φ = `π/4`
Note: when a particle is subjected to two simple harmonic motions at right angle to each other the particle may move along different paths. Such paths are called Lissajous figures.
Displacement vs. time curve for a particle executing S.H.M. is shown in figure. Choose the correct statements.
- Phase of the oscillator is same at t = 0 s and t = 2s.
- Phase of the oscillator is same at t = 2 s and t = 6s.
- Phase of the oscillator is same at t = 1 s and t = 7s.
- Phase of the oscillator is same at t = 1 s and t = 5s.
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)
Which of the following expressions corresponds to simple harmonic motion along a straight line, where x is the displacement and a, b, and c are positive constants?
