Advertisements
Advertisements
Question
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
Solution
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
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.
A small creature moves with constant speed in a vertical circle on a bright day. Does its shadow formed by the sun on a horizontal plane move in a sample harmonic motion?
A pendulum clock gives correct time at the equator. Will it gain time or loose time as it is taken to the poles?
The time period of a particle in simple harmonic motion is equal to the time between consecutive appearances of the particle at a particular point in its motion. This point is
The time period of a particle in simple harmonic motion is equal to the smallest time between the particle acquiring a particular velocity \[\vec{v}\] . The value of v is
A particle moves on the X-axis according to the equation x = A + B sin ωt. The motion is simple harmonic with amplitude
Figure represents two simple harmonic motions.
The parameter which has different values in the two motions is
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 positive in a simple harmonic motion?
The angle made by the string of a simple pendulum with the vertical depends on time as \[\theta = \frac{\pi}{90} \sin \left[ \left( \pi s^{- 1} \right)t \right]\] .Find the length of the pendulum if g = π2 m2.
A simple pendulum of length 1 feet suspended from the ceiling of an elevator takes π/3 seconds to complete one oscillation. Find the acceleration of the elevator.
A simple pendulum of length l is suspended from the ceiling of a car moving with a speed v on a circular horizontal road of radius r. (a) Find the tension in the string when it is at rest with respect to the car. (b) Find the time period of small oscillation.
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
State the laws of the simple pendulum?
Describe Simple Harmonic Motion as a projection of uniform circular motion.
A body oscillates with SHM according to the equation x = 5 cos `(2π"t" + π/4)`. Its instantaneous displacement at t = 1 sec is:
What is the ratio of maxmimum acceleration to the maximum velocity of a simple harmonic oscillator?
A container consist of hemispherical shell of radius 'r ' and cylindrical shell of height 'h' radius of same material and thickness. The maximum value h/r so that container remain stable equilibrium in the position shown (neglect friction) is ______.
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?
