Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Physics for K-12 » Non – uniform circular motion(application)

Navigation

Table of Contents

Recently Viewed

This feature requires Javascript to be enabled.
 

Non – uniform circular motion(application)

Module by: Sunil Kumar Singh. E-mail the author

Summary: Solving problems is an essential part of the understanding process.

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

Hints on problem solving

1: Calculation of acceleration as time rate of change of speed gives tangential acceleration.

2: Calculation of acceleration as time rate of change of velocity gives total acceleration.

3: Tangential acceleration is component of total acceleration along the direction of velocity. Centripetal acceleration is component of acceleration along the radial direction.

4: We exchange between linear and angular quantities by using radius of circle, "r", as multiplication factor. It is helpful to think that linear quantities are bigger than angular quantities. As such, we need to multiply angular quantity by “r” to get corresponding linear quantities and divide a linear quantity by “r” to get corresponding angular quantity.

Representative problems and their solutions

We discuss problems, which highlight certain aspects of the study leading to non-uniform circular motion. The questions are categorized in terms of the characterizing features of the subject matter :

  • Velocity
  • Average total acceleration
  • Total acceleration

Velocity

Example 1

Problem : A particle, tied to a string, starts moving along a horizontal circle of diameter 2 m, with zero angular velocity and a tangential acceleration given by " 4 t 4 t ". If the string breaks off at t = 5 s, then find the speed of the particle with which it flies off the circular path.

Solution : Here, tangential acceleration of the particle at time "t" is given as :

a T = 4 t a T = 4 t

We note here that an expression of tangential acceleration (an higher attribute) is given and we are required to find lower order attribute i.e. linear speed. In order to find linear speed, we need to integrate the acceleration function :

a T = đ v đ t = 4 t a T = đ v đ t = 4 t

đ v = 4 t đ t đ v = 4 t đ t

Integrating with appropriate limits, we have :

đ v = 4 t đ t = 4 t đ t đ v = 4 t đ t = 4 t đ t

v f v i = 4 [ t 2 2 ] 0 5 = 4 X 5 2 2 = 50 m s v f v i = 4 [ t 2 2 ] 0 5 = 4 X 5 2 2 = 50 m s

v f 0 = 50 v f 0 = 50

v f = 50 m s v f = 50 m s

Average total acceleration

Example 2

Problem : A particle is executing circular motion. The velocity of the particle changes from (0.1i + 0.2j) m/s to (0.5i + 0.5j) m/s in a period of 1 second. Find the magnitude of average total acceleration.

Solution : The average total acceleration is :

a = Δ v Δ t = ( 0.5 i + 0.5 j ) - ( 0.1 i + 0.2 j ) 1 a = Δ v Δ t = ( 0.5 i + 0.5 j ) - ( 0.1 i + 0.2 j ) 1

a = ( 0.4 i + 0.3 j ) a = ( 0.4 i + 0.3 j )

The magnitude of acceleration is :

a = ( 0.4 2 + 0.3 2 ) = 0.25 = 0.5 m / s a = ( 0.4 2 + 0.3 2 ) = 0.25 = 0.5 m / s

Example 3

Problem : A particle starting with a speed “v” completes half circle in time “t” such that its speed at the end is again “v”. Find the magnitude of average total acceleration.

Solution : Average total acceleration is equal to the ratio of change in velocity and time interval.

a avg = v 2 v 1 Δ t a avg = v 2 v 1 Δ t

From the figure and as given in the question, it is clear that the velocity of the particle has same magnitude but opposite directions.

Figure 1: The speeds of the particle are same at two positions.
Circular motion
 Circular motion  (nucmq1.gif)

v 1 = v v 1 = v

v 2 = - v v 2 = - v

Putting in the expression of average total acceleration, we have :

a avg = v 2 v 1 Δ t = v v t a avg = v 2 v 1 Δ t = v v t

a avg = 2 v t a avg = 2 v t

The magnitude of the average acceleration is :

a avg = 2 v t a avg = 2 v t

Total acceleration

Example 4

Problem : The angular position of a particle (in radian), on circular path of radius 0.5 m, is given by :

θ = - 0.2 t 2 - 0.04 θ = - 0.2 t 2 - 0.04

At t = 1 s, find (i) angular velocity (ii) linear speed (iii) angular acceleration (iv) magnitude of tangential acceleration (v) magnitude of centripetal acceleration and (vi) magnitude of total acceleration.

Solution : Angular velocity is :

ω = đ θ đ t = đ đ t 0.2 t 2 0.04 = 0.2 X 2 t = 0.4 t ω = đ θ đ t = đ đ t 0.2 t 2 0.04 = 0.2 X 2 t = 0.4 t

The angular speed, therefore, is dependent as it is a function in "t". At t = 1 s,

ω = 0.4 rad / s ω = 0.4 rad / s

The magnitude of linear velocity, at t = 1 s, is :

v = ω r = 0.4 X 0.5 = 0.2 m / s v = ω r = 0.4 X 0.5 = 0.2 m / s

Angular acceleration is :

α = đ ω đ t = đ đ t - 0.4 t = 0.4 rad / s 2 α = đ ω đ t = đ đ t - 0.4 t = 0.4 rad / s 2

Clearly, angular acceleration is constant and is independent of time.

The magnitude of tangential acceleration is :

a T = α r = 0.4 X 0.5 = 0.2 m / s 2 a T = α r = 0.4 X 0.5 = 0.2 m / s 2

Tangential acceleration is also constant and is independent of time.

The magnitude of centripetal acceleration is :

a R = ω v = 0.4 t X 0.2 t = 0.08 t 2 a R = ω v = 0.4 t X 0.2 t = 0.08 t 2

At t = 1 s,

a R = 0.08 m / s 2 a R = 0.08 m / s 2

The magnitude of total acceleration, at t =1 s, is :

a = a T 2 + a R 2 a = a T 2 + a R 2

a = { 0.2 2 + 0.08 2 } = 0.215 m / s 2 a = { 0.2 2 + 0.08 2 } = 0.215 m / s 2

Example 5

Problem : The speed (m/s) of a particle, along a circle of radius 4 m, is a function in time, "t" as :

v = t 2 v = t 2

Find the total acceleration of the particle at time, t = 2 s.

Solution : The tangential acceleration of the particle is obtained by differentiating the speed function with respect to time,

a T = đ v đ t = đ đ t t 2 = 2 t a T = đ v đ t = đ đ t t 2 = 2 t

The tangential acceleration at time, t = 2 s, therefore, is :

a T = 2 X 2 = 4 m / s 2 a T = 2 X 2 = 4 m / s 2

The radial acceleration of the particle is given as :

a R = v 2 r a R = v 2 r

In order to evaluate this expression, we need to know the velocity at the given time, t = 2 s :

v = t 2 = 2 2 = 4 m / s v = t 2 = 2 2 = 4 m / s

Putting in the expression of radial acceleration, we have :

a R = v 2 r = 4 2 4 = 4 m / s 2 a R = v 2 r = 4 2 4 = 4 m / s 2

The total acceleration of the particle is :

a = a T 2 + a R 2 = 4 2 + 4 2 = 4 2 m s 2 a = a T 2 + a R 2 = 4 2 + 4 2 = 4 2 m s 2

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks