A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a
constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and
direction of acceleration of the stone ?
Solution and Explanation
Length of the string, l = 80 cm = 0.8 m
Number of revolutions = 14
Time taken = 25 s
Frequency v =\(\frac{\text{Number of revolutions }} {\text{Time taken}} = \frac{14}{25}\) Hz
Angular frequency, ω = 2πν
\(= 2 x\frac{ 22}{27} x\frac{ 14}{ 25} = \frac{88 }{ 25} rad\,\, s^{-1}\)
Centripetal acceleration, ac = ω2 r
= \((\frac{88}{25})^2\) x 0.8= 9.991 m/s2
The direction of centripetal acceleration is always directed along the string, toward the centre, at all points.
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Concepts Used:
Uniform Circular Motion
A circular motion is defined as the movement of a body that follows a circular route. The motion of a body going at a constant speed along a circular path is known as uniform circular motion. The velocity varies while the speed of the body in uniform circular motion remains constant.
Uniform Circular Motion Examples:
- The motion of electrons around its nucleus.
- The motion of blades of the windmills.
Uniform Circular Motion Formula:
When the radius of the circular path is R, and the magnitude of the velocity of the object is V. Then, the radial acceleration of the object is:
arad = v2/R
Similarly, this radial acceleration is always perpendicular to the velocity direction. Its SI unit is m2s−2.
The radial acceleration can be mathematically written using the period of the motion i.e. T. This period T is the volume of time taken to complete a revolution. Its unit is measurable in seconds.
When angular velocity changes in a unit of time, it is a radial acceleration.
Angular acceleration indicates the time rate of change of angular velocity and is usually denoted by α and is expressed in radians per second. Moreover, the angular acceleration is constant and does not depend on the time variable as it varies linearly with time. Angular Acceleration is also called Rotational Acceleration.
Angular acceleration is a vector quantity, meaning it has magnitude and direction. The direction of angular acceleration is perpendicular to the plane of rotation.
Formula Of Angular Acceleration
The formula of angular acceleration can be given in three different ways.
α = dωdt
Where,
ω → Angular speed
t → Time
α = d2θdt2
Where,
θ → Angle of rotation
t → Time
Average angular acceleration can be calculated by the formula below. This formula comes in handy when angular acceleration is not constant and changes with time.
αavg = ω2 - ω1t2 - t1
Where,
ω1 → Initial angular speed
ω2 → Final angular speed
t1 → Starting time
t2 → Ending time
Also Read: Angular Motion