Examples of linear velocity in the following topics:
-
- Angular velocity ω is analogous to linear velocity v.
- To find the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD.
- This pit moves an arc length Δs in a time Δt, and so it has a linear velocity v = Δs/Δt.
- Similarly, a larger-radius tire rotating at the same angular velocity (ω) will produce a greater linear speed (v) for the car.
- Thus the car moves forward at linear velocity v=rω, where r is the tire radius.
-
- Rolling without slipping can be better understood by breaking it down into two different motions: 1) Motion of the center of mass, with linear velocity v (translational motion); and 2) rotational motion around its center, with angular velocity w.
- The object's angular velocity $\omega$ is directly proportional to its velocity, because as we know, the faster we are driving in a car, the faster the wheels have to turn.
- So, to determine the exact relationship between linear velocity and angular velocity, we can imagine the scenario in which the wheel travels a distance of x while rotating through an angle θ.
- where $dx/dt$ is equal to the linear velocity $v$, and dθ/dt is equal to the angular velocity $\omega$.
-
- The familiar linear vector quantities such as velocity and momentum have analogous angular quantities used to describe circular motion.
- Linear motion is motion in a straight line.
- This type of motion has several familiar vector quantities associated with it, including linear velocity and momentum.
- We recall from our study of linear velocity that a change in the direction of the velocity vector, is a change in velocity and a change in velocity is acceleration.
- Just like with linear acceleration, angular acceleration is a change in the angular velocity vector.
-
- As mentioned in previous sections on kinematics, any change in velocity is given by an acceleration.
- Often the changes in velocity are changes in magnitude.
- When an object speeds up or slows down this is a change in the objects velocity.
- However, because velocity is a vector, it also has a direction.
- where $v$ is the linear velocity of the object and $r$ is the radius of the circle.
-
- The description of circular motion is described better in terms of angular quantity than its linear counter part.
- Here, the velocity of particle is changing - though the motion is "uniform".
- The velocity (i.e. angular velocity) is indeed constant.
- Second advantage is that angular velocity conveys the physical sense of the rotation of the particle as against linear velocity, which indicates translational motion.
- As we use mass, linear momentum, translational kinetic energy, and Newton's 2nd law to describe linear motion, we can describe a general rotational motion using corresponding scalar/vector/tensor quantities:
-
- Angular acceleration gives the rate of change of angular velocity.
- When the axis of rotation is perpendicular to the position vector, the angular velocity may be calculated by taking the linear velocity $v$ of a point on the edge of the rotating object and dividing by the radius .
- A fly on the edge of a rotating object records a constant velocity $v$.
- The object is rotating with an angular velocity equal to $\frac{v}{r}$.
- The direction of the angular velocity will be along the axis of rotation.
-
- Calculate from the Euler equation and the continuity equation, at what velocity does the flux ($\rho V$) reach its maximum for fluid flowing through a tube of variable cross-sectional area?
- At which velocities does the flux vanish?
- What are the values of the critical velocities $v_0$?
- Show that for a linear sound wave i.e. one in which $\delta \rho \ll \rho$ that the velocity $v$ of fluid motion is much less than $c_s$.
- Estimate the maximum longitudinal fluid velocity in the case of a sound wave in air at STP in the case of a disturbance which sets up pressure fluctuations of order 0.1\%.
-
- Constant angular acceleration describes the relationships among angular velocity, angle of rotation, and time.
- We have already studied kinematic equations governing linear motion under constant acceleration:
- To determine this equation, we use the corresponding equation for linear motion:
- As in linear kinematics where we assumed a is constant, here we assume that angular acceleration α is a constant, and can use the relation: $a=r\alpha $ Where r - radius of curve.Similarly, we have the following relationships between linear and angular values: $v=r\omega \\x=r\theta $
- Relate angle of rotation, angular velocity, and angular acceleration to their equivalents in linear kinematics
-
- In a closed system, angular momentum is conserved in a similar fashion as linear momentum.
- This fact is readily seen in linear motion.
- When an object of mass m and velocity v collides with another object of mass m2 and velocity v2, the net momentum after the collision, mv1f + mv2f, is the same as the momentum before the collision, mv1i + mv2i.
- This equation is an analog to the definition of linear momentum as p=mv.
- An object that has a large angular velocity ω, such as a centrifuge, also has a rather large angular momentum.
-
- Angular acceleration is the rate of change of angular velocity.
- Angular acceleration is defined as the rate of change of angular velocity.
- where $\Delta \omega$ is the change in angular velocity and $\Delta t$ is the change in time.
- It is useful to know how linear and angular acceleration are related.
- Tangential acceleration refers to changes in the magnitude of velocity but not its direction.