Units of Rotary Motion
Varieties of Rotational Movement
The three categories of Rotational Motion are as follows:
- Pure Rotation, also referred to as Motion about a fixed axis, characterizes an object’s rotation around a stationary point. Instances include the rotation of a ceiling fan’s blades or the rotation of the hands on an analog clock, both of which revolve around a central fixed point.
- A combination of Rotational and Translational Motion. This type of movement describes an object whose components can rotate around a stationary point while the object moves in a straight line. The rolling of car wheels is an example of this. The wheels have two velocities, one from the rotation of the wheel and the other from the car’s translational motion.
- Rotation around an Axis of Rotation. This type of motion describes objects that rotate around an axis while also rotating around another object. The Earth orbiting the sun while simultaneously rotating around its own axis is an example.
Physics of Rotational Movement
The principles that govern the motion of rotating objects fall under the domain of kinematics in physics. Kinematics is a branch of physics that concentrates on the motion of an object without considering the forces that cause it. Kinematics examines quantities such as acceleration, velocity, displacement, and time that can be related to both linear and rotational motion. In the study of rotational motion, we employ the principles of rotational kinematics, which explore the connection between rotational motion variables.
Variables of Rotational Movement
The quantities that describe rotational motion include:
- Angular velocity
- Angular acceleration
- Angular displacement
- Time
Correlation Between Rotational Kinematics and Linear Kinematics
Before delving into rotational kinematics, it’s vital to comprehend the association between kinematic variables. This relationship can be observed by examining the variables listed in the table below.
Rotational Kinematics and Dynamics
In addition to rotational kinematics, it is crucial to consider rotational dynamics, which addresses an object’s motion and the forces that cause it to rotate. In rotational motion, torque is the force responsible for rotation.
Newton’s Second Law of Rotational Motion
We will now define torque and its corresponding mathematical equation below.
Definition of Torque
To express Newton’s second law in the context of rotational motion, it is necessary to provide a definition of torque.
Rotational dynamics examines the motion of objects in rotation and explores how forces affect this motion.
Rotational Motion of a Point Particle
Units for the Following Variables (all units see here):
m is denotes the mass of a particle moving in the x-y plane
represents the force vector applied within the plane of motion
represents the velocity vector that is tangent to the particle’s trajectory
represents the linear momentum vector, which is parallel to the velocity vector
represents the radius vector of curvature of the particle’s trajectory, which is perpendicular to the trajectory
represents the angular velocity vector, which is perpendicular to the plane of motion
represents the angular acceleration vector, which is perpendicular to the plane of motion
represents the angular momentum, which is parallel to the angular velocity vector
represents the torque associated with the force, which is perpendicular to the plane of motion
, normal to plane of motion d denotes the lever arm length
Formulas for General Cases
Formula for the Moment of Inertia of a Particle about the center of rotation
Definition of the Angular Momentum Vector using vector product
where is the linear momentum vector that is perpendicular to
Relationship between the Angular Momentum and Angular Velocity Vectors
Calculation of the Magnitude of the Angular Momentum
Definition of the Torque Vector using vector product
Calculation of the Magnitude of the Torque
where:
is the angle between the vectors
and
, shown in the above diagram
is level arm (or moment arm) of
Angular Form of Newton’s Second Law:
– for general case – for constant moment of inertia
Rotation of a Symmetric Solid around its Axis of Symmetry
The moment of inertia of a symmetric solid around its axis of rotation can be calculated using the formula:
where
mi represents a small portion of mass number i located at distance Ri between its center and axis of rotation (for i = 1, 2, 3, … , n)
dV is the infinitesimal volume with density at distance R from axis of rotation
Parallel Axis Theorem
where:
I is moment of inertia of solid of mass m about axis located at distance l from its center of mass
Icm is moment of inertia of the solid about axis passing thought the ceneter of mass and parallel the the previous axis
The angular momentum of a symmetric solid rotating around its axis of symmetry is given by the formula:
where is angular velocity of the solid
Newton’s Second Law in angular form:
– for general case
– for constant moment of inertia where
is net torque about axis of rotation associated with net external force
The Rotation of a System Of Particles in the General Case
The total angular momentum vector of the system with respect to an arbitrary point C
where and
are position vector and linear momentum vector for i-th particle with respect to the point C (for i = 1, 2, 3, …, n)
The total torque about point C resulting from external forces
where is external force applied at point
with respect to the point C (for j = 1, 2, 3, …, k)
Newton’s Second Law in angular form
The law of conservation of angular momentum of the system
If then
about point C
Motion of Spinning Top Under Gyroscopic Effect
The following variables are defined:
is the angular velocity of the spinning top about its axis
is a vertical external force applied to the top
is the radius-vector from the axis of the spinning top to the point where the force
is applied to the top
is the precessional frequency of the top about the z-axis
Equation of motion for the spinning top is given by:
where I is the moment of inertia of the top about its axis
The precessional frequency is determined by the expression:
Angular velocity is the rate at which an object rotates or moves around a central axis. It is usually measured in radians per second (rad/s) and is equal to the change in angle (in radians) over time.
Torque is the rotational equivalent of force. It is a measure of the twisting force that causes an object to rotate around a central axis. Torque is usually measured in newton-meters (N⋅m).
Rotational inertia (also known as moment of inertia) is a measure of an object’s resistance to changes in rotational motion. It depends on the mass of the object and the distribution of that mass around the axis of rotation.
Angular velocity is calculated by dividing the change in angle (in radians) by the change in time. The formula is: angular velocity = Δθ/Δt.
Torque is calculated by multiplying the force applied to an object by the distance between the axis of rotation and the point where the force is applied. The formula is: torque = force × distance.
Rotational inertia depends on the mass of the object and the distribution of that mass around the axis of rotation. The formula for rotational inertia depends on the shape of the object. For example, the rotational inertia of a solid cylinder is 1/2 × mass × radius^2.
The units of angular velocity are radians per second (rad/s).
The units of torque are newton-meters (N⋅m).
The units of rotational inertia depend on the units used to measure mass and distance. For example, if mass is measured in kilograms and distance is measured in meters, the units of rotational inertia would be kilograms times meters squared (kg⋅m^2).
These concepts are applicable in many real-world scenarios, such as the design of engines, turbines, and other rotating machinery. They are also important in understanding the behavior of objects in motion, such as the movement of planets around the sun. By mastering these concepts, you can gain a deeper understanding of the physical world around us.
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