Uniform Circular Motion - Definition, Examples, Diagram, Characteristic, Examples, FAQs
An object is said to be in a uniform circular motion when it travels a circular path at constant speed. This means that equal distances along the circular path are traveled in equal intervals of time; however, the direction of motion is continuously changing. In this article, we shall study the complex mechanism of uniform circular motion: what forces allow for circular motion and how velocity and acceleration apply in this context.
- What Is Uniform Circular Motion?
- Uniform Circular Motion Formula
- Terms Related to Uniform Circular Motion
- Relation Between Angular Acceleration and Tangential Acceleration
- Uniform Circular Motion Questions and Answers
This concept fits into the greater picture of kinematics, which is very important in Class 11 physics. Thus, it is important for board exams and competitive examinations conducted by different national authorities such as the Joint Entrance Examinations (JEE Main) and the National Eligibility Entrance Test (NEET), apart from others like SRMJEE, BITSAT, WBJEE, and BCECE. The concept has been asked nine times over the past ten years in the JEE Main examination. For NEET, five questions were asked about this concept.
What Is Uniform Circular Motion?
Circular motion is the motion of a body travelling in a path around a fixed point in the shape of a circle. Uniform circular motion implies that if the body travels equal distances along the circumference of the circle in equal intervals of time, then the motion is said to be uniform circular motion. Naturally, this means that it is a type of circular motion in which the speed is maintained while the direction of the velocity is altered.
Uniform circular motion examples:
(i) Motion of the moon around the Earth
(ii) Motion of a satellite around its planet
Note: Circular motion is also known as accelerated motion
Uniform Circular Motion Formula
Uniform circular motion Formula Class 9 or other classes are given here.
In uniform circular motion, a particle moves with constant speed.
- Angular displacement $\Delta \theta=\frac{\operatorname{Arc}\left(P P^{\prime}\right)}{r}$
- Angular velocity $\omega=\frac{\Delta \theta}{\Delta t}=\frac{2 \pi}{T}=2 \pi v$
- Linear speed $v=r_\omega$
Centripetal acceleration is due to a change in direction of velocity and is always directed towards the centre.
$
a={v^2/r}=r \omega^2=4 \pi^2 v^2 r=v \omega
$
Terms Related to Uniform Circular Motion
The term related to circular motion is given below:
Radius vector
The vector joining the centre of the circular path to the position on the circular path is called the radius vector.
Angular position
The angle made by the radius vector with a reference line (arbitrarily chosen diameter) is called angular position. The direction of angular position can be clockwise or anticlockwise depending upon the choice of frame of reference. The angular position of the particle at position "P" is denoted by the y $\text { angle } \theta$ in the diagram above.
Angular displacement
The change in angular position is called angular displacement. It is the angle through which the radius vector rotates during the given circular motion.
The angular displacement between positions 'P' and 'Q' is denoted $\text { by } \Delta \theta$ in the diagram above with the S.I unit of angular position and angular displacement is Radian.
Angular velocity
Denoted by $\omega$ (omega)
$\omega$-Rate of change of angular displacement.
Average angular velocity-
$
\omega_{\text {avg }}=\frac{\Delta \theta}{\Delta t}
$
Instantaneous angular velocity
$
\omega=\frac{d \theta}{d t}
$
S.I. units- Radian per second (rad per sec )
$\omega$ is a vector quantity
The direction of $\omega$ is given by the Right-hand rule.
According to the right-hand rule, if you hold the axis with your right hand and rotate the fingers in the direction of motion of the rotating body, then the thumb will point in the direction of the angular velocity.
Angular Acceleration
The rate of change of angular velocity with time is said to be Angular Acceleration.
$
\alpha=\frac{\Delta \omega}{\Delta t}
$
- SI units- $\operatorname{rad} .(\mathrm{sec})^{-2}$
- Angular Acceleration is a vector quantity.
The direction of Angular Acceleration
a) If angular velocity is increasing then the direction of Angular Acceleration is in the direction of angular velocity.
b) If angular velocity is decreasing then the direction of Angular Acceleration is in the direction which is opposite to the direction of angular velocity.
Time is taken to complete one rotation
Formula-
$
T=\frac{2 \pi}{\omega}
$
Where $\omega=$ angular velocity
If $\mathrm{N}=\mathrm{no}$. of revolutions and total time then
$
T=\frac{t}{N} \text { or } \quad\left(\omega=\frac{2 \pi N}{t}\right)
$
- S.I unit - seconds (s)
Frequency
The total number of rotations in one second.
Formula-
$
\nu=\frac{1}{T}
$
- S.I. unit $=$ Hertz
We can write the relation between angular frequency and frequency as
$
w=2 \pi \nu
$
Centripetal Acceleration and Tangential acceleration
Centripetal acceleration: When a body is moving in a uniform circular motion, a force is responsible for changing the direction of its velocity. This force acts towards the centre of the circle and is called centripetal forceThe acceleration produced by this force is centripetal acceleration.
Formula-
$
a_c=\frac{V^2}{r}
$
Where $a_c=$ Centripetal acceleration,
$\mathrm{V}=$ linear velocity
$r=$ radius
Tangential acceleration: During circular motion, if the speed is not constant, then along with centripetal acceleration, there is also a tangential acceleration, which is equal to the rate of change of magnitude of linear velocity.
$a_t=\frac{\mathrm{d} v}{\mathrm{~d} t}$
Relation Between Angular Acceleration and Tangential Acceleration
$\overrightarrow{a_t}=\vec{\alpha} \times \vec{r}$
Where:
- $\overrightarrow{a_t}=$ Tangential acceleration (linear acceleration along the tangent to the circular path)
- $\vec{\alpha}=$ Angular acceleration vector (rate of change of angular velocity)
- $\vec{r}^*=$ Position vector (radius vector from axis of rotation to the point of interest)
Total acceleration
The vector sum of Centripetal acceleration and tangential acceleration is called Total acceleration.
Formula-
$a_n=\sqrt{a_c^2+a_t^2}$
The angle between Net acceleration and tangential acceleration $(\theta)$
From the above diagram-
$
\tan \theta=\frac{a_c}{a_t}
$
Also Check-
| Projectile Motion |
| Horizontal Projectile Motion |
| Equation Of the Path Of A Projectile |
| Relative velocity |
| Boat River Problem |
Uniform Circular Motion Questions and Answers
Example 1: If a body moving in a circular path maintains a constant speed of 10 ms-1, then which of the following correctly describes the relation between acceleration and radius?
1) 
2) 
3) 
4) 
Solution:
$a=\frac{v^2}{r}$
Figure Shows Centripetal acceleration
$\begin{aligned}
& a=\frac{v^2}{r} \because|\vec{v}|=\text { constant } \\
& a \propto \frac{1}{r} \text { or } a r=\text { constant }
\end{aligned}$
Hence, the graph between a and r will be a hyperbola.
Example 2: A Point P moves in a counter-clockwise direction on a circular path as shown in the figure. The movement of $P$ is such that it sweeps out a length that is in metres and $t$ is in seconds. The radius of the path is $\mathbf{2 0 ~ m}$, The acceleration (in $\mathrm{m} / \mathrm{s}^2$ ) of $P$ When $t=2 s$ is nearly
1) 14
2) 13
3) 12
4) 7.2
Solution:
$
\begin{aligned}
& \text { As } S=t^3+3 \\
& V=\frac{d s}{d t}=3 t^2+0 \\
& \Rightarrow v=3 t^2
\end{aligned}
$
tangential acceleration
$
\begin{aligned}
& =a_t=\frac{d v}{d t}=\frac{d\left(3 t^2\right)}{d t} \\
& a_t=6 t
\end{aligned}
$
At $t=2 \mathrm{sec}$
$
\begin{aligned}
& v=3(2)^2=12 \mathrm{~ms}^{-1} \\
& a_t=6 \times 2=12 \mathrm{~ms}^{-2}
\end{aligned}
$
$
\begin{aligned}
& \therefore \text { centripetal acceleration }=\vec{a}_c=\frac{v^2}{r}=\frac{(12)^2}{20}=\frac{144}{20} \\
& a_c=7.2 \mathrm{~ms}^{-2}
\end{aligned}
$
$\therefore$ Net acceleration
$
\begin{aligned}
& a=\sqrt{a_c^2+a_t^2}=\sqrt{7.2^2+12^2} \\
& a_c \simeq 14 \mathrm{~ms}^{-2}
\end{aligned}
$
Example 3: A particle is moving with speed varying as v = 2t, then the angle which the resultant acceleration makes with the radial direction (R=1m) at t = 2 is
$\begin{aligned} & \text { 1) } \tan ^{-1}\left(\frac{1}{2}\right) \\ & \text { 2) } \tan ^{-1}\left(\frac{1}{6}\right) \\ & \text { 3) } \tan ^{-1}\left(\frac{1}{8}\right) \\ & \text { 4) } \tan ^{-1}\left(\frac{1}{4}\right)\end{aligned}$
Solution:
The angle between Total acceleration and centripetal acceleration is given by
$
\tan \phi=\frac{a_t}{a_c}=\frac{r^2 \alpha}{V^2}
$
where
$\alpha=$ angular acceleration
$V=$ velocity
$r=$ radius of circle
So From the below figure
$\begin{aligned}
& \tan \theta=\frac{a_t}{a_r} \\
& a_t=\frac{\mathrm{d} v}{\mathrm{~d} t}=2 \mathrm{~m} / \mathrm{s}^2 \\
& a_r=\frac{v^2}{R} \\
& a_r=\frac{4 t^2}{1}=4 \times 2^2=16 \\
& \therefore \tan \theta=\frac{2}{16}=\frac{1}{8} \\
& \therefore \theta=\tan ^{-1}\left(\frac{1}{8}\right)
\end{aligned}$
Example 4: A particle is moving with a constant speed of 8 m/s in a circular path of radius 1 m. What will be the displacement of the particle in 1 sec?
1) 2 sin 80
2) 2 sin 40
3) 4 sin 80
4) 4 sin 40
Solution:
Displacement in Circular Motion -
$
\Delta r=2 r \sin \frac{\theta}{2}
$
$\Delta r=$ displacement
$\theta=$ Angle between two vectors
- wherein
$
\text { If }\left|\overrightarrow{r_1}\right|=\left|\overrightarrow{r_2}\right|=r
$
Let the angular displacement of the particle from $\mathrm{A}$ to $\mathrm{B}$ will be
$
2 R \sin \frac{\Theta}{2}
$
Length of circular arc $A B=8 * 1=8 \mathrm{~m}$
$
\begin{aligned}
& \text { Angle } \Theta=\frac{\text { arc length }}{\text { Radius }}=\frac{\Theta}{1}=8 \mathrm{rad} \\
& d=2 R \sin \frac{\Theta}{2}=2 * 1 * \sin \frac{8}{2} \\
& 2 \sin 4^0
\end{aligned}
$
Hence, the correct answer is option (2).
Uniform circular motion involves a particle moving with constant speed along a circular path, whose velocity continuously changes due to the continuous change in direction. A simple example is the merry-go-round in which objects move in circles at an even speed. This concept also applies to natural phenomena like planets orbiting the sun. This understanding of uniform circular motion allows us to comprehend how forces and motion work in circular paths, which is essential for our daily lives and for grasping bigger movements in cosmology.
Related Study Resource,
Frequently Asked Questions (FAQs)
In physics, when a body moves in any circular path but of fixed radius and moves with constant value of speed, then the motion performed by such bodies are referred to as uniform circular motion. Examples of uniform circular motion are such as: a boy moving in circular ground with constant speed, the motion of atomic particles such as electrons in an atom in its orbit is also an example of uniform circular motion.
Uniform circular motion is the movement of an object in a circular path at a constant speed. The object's direction constantly changes, but its speed remains the same. This type of motion is characterized by a constant angular velocity and a constant radius from the center of the circle.
According to definition, a particle performing uniform circular motion must have fixed radius of circular path and it must move with constant speed, so we have given that particle is moving in circular path having fixed radius of r=5m and constant speed v=2ms-1. So, yes, particles are performing uniform circular motion. Acceleration of uniform circular motion is given by a=v2r on putting the values, we get, acceleration of uniform circular motion is a=45=0.8ms-2.
According to the definition of uniform circular motion, if a body is moving in any circular path and moving with uniform speed then, motion performed by the body is known as uniform circular motion. Hence, the correct option is (D) Uniform circular motion.
For a particle performing uniform circular motion, it must have constant speed and constant radius and the time period of uniform circular motion is also constant. It’s only velocity which keeps changing its direction at every instant of motion. So correct options are (B), (C), and (D).
The most important point of a particle performing uniform circular motion is that the magnitude of speed is always constant. but remember, the velocity changes at every instant of motion as velocity is a vector quantity and the direction of velocity is tangential to the point on circular path, which keeps changing at every instant, So, the fixed value of speed of a particle moving in circular path is the important characteristic of uniform circular motion.
Rotational inertia, also known as moment of inertia, plays a crucial role in uniform circular motion, especially for extended objects. While point masses in uniform circular motion are often considered for simplicity, real objects have distributed mass. The rotational inertia affects how easily an object can be set into or taken out of circular motion. Objects with greater rotational inertia require more torque to achieve the same angular acceleration, which is important in understanding the dynamics of rotating systems.
In uniform circular motion, angular momentum is conserved as long as no external torque is applied. The angular momentum (L) is given by L = Iω, where I is the moment of inertia and ω is the angular velocity. For a point mass in circular motion, L = mvr, where m is the mass, v is the velocity, and r is the radius. The conservation of angular momentum is why ice skaters spin faster when they pull their arms in, reducing their moment of inertia.
Uniform circular motion is closely related to wave motion, particularly in the context of simple harmonic motion. The projection of uniform circular motion onto a straight line results in simple harmonic motion, which is the basis for many types of waves. This relationship helps in understanding concepts like phase, amplitude, and frequency in wave motion, and is fundamental to the study of oscillations and waves in physics.
In uniform circular motion, the velocity vector is always tangent to the circular path, with no radial component. The acceleration, however, has only a radial component (the centripetal acceleration) and no tangential component. This radial acceleration constantly changes the direction of the velocity vector without changing its magnitude, resulting in the circular path.
An object in uniform circular motion experiences acceleration because its velocity is constantly changing in direction, even though its speed remains constant. This change in velocity (direction) results in centripetal acceleration, which is always directed towards the center of the circular path.
The force responsible for keeping an object in uniform circular motion is called the centripetal force. This force is always directed towards the center of the circular path and causes the object to continually change its direction, maintaining the circular motion.
The centripetal acceleration (a) in uniform circular motion is inversely proportional to the radius (r) of the circular path, given by the equation a = v²/r, where v is the linear velocity. This means that for a given speed, a smaller radius results in a larger centripetal acceleration.
Friction plays a crucial role in uniform circular motion on a flat, horizontal surface by providing the necessary centripetal force. Without friction, an object would move in a straight line instead of following a circular path. The static friction between the object and the surface acts as the centripetal force.
Angular velocity (ω) is the rate of change of angular position, measured in radians per second. Linear velocity (v) is the tangential speed of the object, measured in distance per unit time. They are related by the equation v = ωr, where r is the radius of the circular path.
The period (T) of uniform circular motion is inversely related to its frequency (f). The relationship is expressed as T = 1/f. This means that as the frequency increases, the period decreases, and vice versa.
Uniform circular motion involves constant speed and a fixed radius, resulting in constant angular velocity. Non-uniform circular motion can involve changes in speed, radius, or both, leading to varying angular velocity. In uniform circular motion, only the direction of velocity changes, while in non-uniform circular motion, both the magnitude and direction of velocity can change.
The relationship between the angle traversed (θ) and the arc length (s) in uniform circular motion is given by the equation s = rθ, where r is the radius of the circular path. This equation holds true when θ is measured in radians. It shows that the arc length is directly proportional to both the radius and the angle traversed.
Uniform circular motion and simple harmonic motion are closely related. The projection of uniform circular motion onto a diameter of the circle results in simple harmonic motion. This relationship is fundamental in understanding oscillations and waves, as many harmonic oscillators can be modeled using this connection.
The tangential velocity in uniform circular motion represents the instantaneous linear velocity of the object at any point on its circular path. It is always perpendicular to the radius and tangent to the circle. The magnitude of this velocity remains constant in uniform circular motion, while its direction continuously changes.
The Earth's rotation is an example of uniform circular motion on a large scale. Each point on the Earth's surface undergoes circular motion as the planet rotates on its axis. The speed of this motion varies with latitude, being greatest at the equator and zero at the poles. This rotation is responsible for the day-night cycle and influences various phenomena like the Coriolis effect.
In a conical pendulum (where a mass swings in a horizontal circle at the end of a string), the tension in the string provides the centripetal force necessary for uniform circular motion. The tension is directed along the string and has a component towards the center of the circle. This tension, along with the weight of the mass, determines the angle of the pendulum and its period of rotation.
Centrifuges utilize the principles of uniform circular motion to separate substances of different densities. As the centrifuge rotates, the samples inside experience a centripetal force. This force causes denser substances to move farther from the axis of rotation, while less dense substances move closer to the center. The separation is based on the fact that the centripetal force increases with distance from the center of rotation.
Geostationary satellites utilize the principles of uniform circular motion to maintain a fixed position relative to Earth. These satellites orbit in the equatorial plane with a period equal to Earth's rotational period (24 hours). The satellite's altitude is carefully chosen so that its orbital velocity matches Earth's rotational velocity, allowing it to appear stationary from Earth's surface.
Particle accelerators, such as cyclotrons and synchrotrons, use principles of uniform circular motion to accelerate charged particles. In these devices, magnetic fields provide the centripetal force necessary to keep particles moving in a circular path. As particles gain energy and speed, the magnetic field strength or the radius of the path is adjusted to maintain the circular motion, allowing particles to be accelerated to very high energies.
The Coriolis effect, while not directly a result of uniform circular motion, is related to the rotation of the Earth, which is a form of circular motion. This effect causes moving objects on Earth to appear to deflect from their path when viewed from the rotating Earth's reference frame. Understanding circular motion is crucial to comprehending how the Earth's rotation influences the apparent path of moving objects on its surface.
A centrifugal governor uses the principles of uniform circular motion to regulate the speed of engines. As the engine speed increases, the rotating masses of the governor move outward due to the increased centrifugal effect. This outward motion is used to control the engine's power supply, typically by adjusting a valve, thus maintaining a constant speed. This application demonstrates how the forces in circular motion can be harnessed for practical purposes.
The mass of an object does not affect its uniform circular motion in terms of its speed or radius. However, it does affect the amount of centripetal force required to maintain the circular motion. A more massive object requires a greater centripetal force to maintain the same circular path at the same speed.
Centrifugal force is not a real force but rather a fictitious or apparent force perceived in a rotating reference frame. In uniform circular motion, an observer in the rotating frame might perceive an outward force (centrifugal), but this is actually the result of the object's inertia resisting the centripetal force that keeps it in circular motion.
For satellites orbiting the Earth, the centripetal force necessary for their circular (or nearly circular) orbit is provided by Earth's gravitational attraction. The satellite's velocity and altitude are carefully calculated so that the centripetal force required for the orbit exactly matches the gravitational force exerted by Earth, keeping the satellite in a stable orbit.
The formula F = mv²/r is crucial in uniform circular motion as it relates the centripetal force (F) to the mass of the object (m), its velocity (v), and the radius of the circular path (r). This formula shows that the force required increases with mass and velocity but decreases with a larger radius. It's essential for calculating the force needed to maintain circular motion or determining the resulting motion given a specific force.
In uniform circular motion on a vertical circle (like a roller coaster loop), the normal force varies at different points of the circle. At the bottom, it's greatest, as it must overcome both gravity and provide the centripetal force. At the top, it's least, as gravity assists in providing the centripetal force. The normal force, along with gravity, contributes to the centripetal force needed for the circular motion.
While planetary orbits are not perfectly circular, the concept of uniform circular motion provides a simplified model for understanding planetary motion. Planets move in elliptical orbits, but circular approximations can be useful for basic calculations. The gravitational force from the sun acts as the centripetal force keeping planets in their orbits.
Banking curves, such as those on highways or racetracks, use the principles of uniform circular motion. The road is tilted inward, creating an angle with the horizontal. This banking provides a component of the normal force that acts as the centripetal force, allowing vehicles to navigate the curve at higher speeds without relying solely on friction.
Air resistance can affect uniform circular motion by opposing the motion of the object. In ideal scenarios, air resistance is often neglected. However, in real-world situations, especially at high speeds, air resistance can cause a gradual decrease in velocity, potentially turning uniform circular motion into a spiral motion as the object slows down and its path changes.
The critical velocity in vertical circular motion is the minimum velocity required for an object to complete a vertical circle without falling, particularly at the top of the circle. At this point, the centripetal force is provided solely by gravity. If the velocity is less than the critical velocity, the object will not complete the circle and will fall.
Centripetal force (F) and centripetal acceleration (a) are directly related through Newton's Second Law of Motion: F = ma, where m is the mass of the object. The centripetal force causes the centripetal acceleration, which is always directed towards the center of the circular path.
While the actual motion of electrons in atoms is more complex and governed by quantum mechanics, the Bohr model of the atom uses the concept of uniform circular motion as a simplified representation. In this model, electrons are thought to orbit the nucleus in circular paths, with the electrostatic force acting as the centripetal force. This simplified model helps in understanding basic atomic structure and energy levels.
Angular displacement is the angle through which an object rotates in circular motion, measured in radians or degrees. Linear displacement is the straight-line distance between the initial and final positions of the object. In uniform circular motion, the angular displacement increases continuously, while the linear displacement oscillates and can never exceed the diameter of the circle.
In uniform circular motion, the tangential acceleration is zero because the speed of the object remains constant. All the acceleration is in the radial direction (towards the center), which is the centripetal acceleration. This constant speed but changing direction is what defines uniform circular motion and distinguishes it from other types of circular motion where speed may vary.
The centripetal force is crucial in maintaining uniform circular motion as it constantly pulls the object towards the center of the circle, changing its direction but not its speed. If the centripetal force is suddenly removed, the object will continue moving in a straight line tangent to the circle at the point where the force was removed, following Newton's First Law of Motion.
The ratio v²/r in uniform circular motion represents the centripetal acceleration. This ratio remains constant for a given circular motion, regardless of the mass of the object. It shows that for a fixed radius, the acceleration increases with the square of the velocity, and for a fixed velocity, the acceleration is inversely proportional to the radius. This relationship is crucial in designing safe curves on roads and in understanding planetary orbits.
When a charged particle enters a uniform magnetic field perpendicular to its velocity, it experiences a magnetic force perpendicular to both its velocity and the magnetic field. This force acts as the centripetal force, causing the particle to move in a circular path. The radius of this path depends on the particle's mass, charge, velocity, and the strength of the magnetic field. This principle is used in devices like mass spectrometers and cyclotrons.
The spin cycle in a washing machine utilizes uniform circular motion to remove water from clothes. As the drum rotates at high speed, the clothes are pressed against the drum's perforated walls. The centripetal force keeps the clothes in circular motion, while water, being denser, experiences a greater outward force and is forced through the perforations. This application demonstrates how the principles of circular motion can be used for practical purposes in everyday appliances.
The centripetal force being perpendicular to the velocity in uniform circular motion is crucial because it ensures that the force only changes the direction of motion, not the speed. This perpendicular relationship means that the force does no work on the object, as work is only done when a force has a component parallel to the displacement. Consequently, the kinetic energy of the object remains constant, maintaining the uniform speed characteristic of this type of motion.
Roundabouts in traffic systems are designed based on principles of uniform circular motion. The circular design allows vehicles to maintain a relatively constant speed while changing direction, which is more efficient than coming to a complete stop at an intersection. The radius of the roundabout is calculated to allow for safe navigation at appropriate speeds, considering factors like centripetal force, friction, and typical vehicle capabilities. This application of circular motion principles helps in creating smoother, safer traffic flow.
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