Dynamics

DynamicsNewton’s Laws Of Motion

DynamicsNewton’s Laws Of Motion

Law 1“ Corpus omne perseverare in statu suo quiescendi vel movendi

uniformiter, nisi quatenus a viribus impressis cogitur statum illum mutare.”

DynamicsNewton’s Laws Of Motion

Law 1“ Corpus omne perseverare in statu suo quiescendi vel movendi

uniformiter, nisi quatenus a viribus impressis cogitur statum illum mutare.”

Everybody continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.

DynamicsNewton’s Laws Of Motion

Law 1“ Corpus omne perseverare in statu suo quiescendi vel movendi

uniformiter, nisi quatenus a viribus impressis cogitur statum illum mutare.”

Everybody continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.Law 2“ Mutationem motus proportionalem esse vi motrici impressae, et

fieri secundum rectam qua vis illa impritmitur.”

DynamicsNewton’s Laws Of Motion

Law 1“ Corpus omne perseverare in statu suo quiescendi vel movendi

uniformiter, nisi quatenus a viribus impressis cogitur statum illum mutare.”

Everybody continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.Law 2“ Mutationem motus proportionalem esse vi motrici impressae, et

fieri secundum rectam qua vis illa impritmitur.”

The change of motion is proportional to the motive force impressed, and it is made in the direction of the right line in which that force is impressed.

motionin changeforceMotive

motionin changeforceMotive onacceleratiF

motionin changeforceMotive onacceleratiF

onacceleraticonstant F

motionin changeforceMotive onacceleratiF

maF

onacceleraticonstant F

(constant = mass)

motionin changeforceMotive onacceleratiF

maF

onacceleraticonstant F

(constant = mass)

Law 3“ Actioni contrariam semper et aequalem esse reactionem: sive

corporum duorum actions in se mutuo semper esse aequales et in partes contrarias dirigi.”

motionin changeforceMotive onacceleratiF

maF

onacceleraticonstant F

(constant = mass)

Law 3“ Actioni contrariam semper et aequalem esse reactionem: sive

corporum duorum actions in se mutuo semper esse aequales et in partes contrarias dirigi.”

To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed in contrary parts.

Acceleration

Acceleration

dtdvdt

xdx

2

2

Acceleration

dtdvdt

xdx

2

2

(acceleration is a function of t)

Acceleration

dtdvdt

xdx

2

2

(acceleration is a function of t)

2

21 v

dxd (acceleration is a function of x)

Acceleration

dtdvdt

xdx

2

2

(acceleration is a function of t)

2

21 v

dxd (acceleration is a function of x)

dxdvv (acceleration is a function of v)

Acceleration

dtdvdt

xdx

2

2

(acceleration is a function of t)

2

21 v

dxd (acceleration is a function of x)

dxdvv (acceleration is a function of v)

e.g. (1981) Assume that the earth is a sphere of radius R and that, at a point r from the centre of the earth, the acceleration due to gravity is proportional to and is directed towards the earth’s centre.

A body is projected vertically upwards from the surface of the earth with initial speed V.

R

21r

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0vrm resultant

force

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

2rm

rm resultantforce

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

2rm

rm resultantforce

2rmrm

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

2rm

rm resultantforce

2rmrm

2rr

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

2rm

rm resultantforce

2rmrm

2rr

grRr ,when

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

2rm

rm resultantforce

2rmrm

2rr

grRr ,when

2

2i.e.

gRR

g

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

2rm

rm resultantforce

2rmrm

2rr

grRr ,when

2

2i.e.

gRR

g

2

2

rgRr

(i) Prove that it will escape the earth if and only if where g is the magnitude of the acceleration due to gravity at the earth’s surface.

gRV 2

O

grVvRr ,,

0v

2rm

rm resultantforce

2rmrm

2rr

grRr ,when

2

2i.e.

gRR

g

2

2

rgRr

2

22

21

rgRv

drd

cr

gRv

cr

gRv

22

22

221

cr

gRv

cr

gRv

22

22

221

VvRr ,when

cr

gRv

cr

gRv

22

22

221

VvRr ,when

gRVc

cRgRV

2

2 i.e.

2

22

cr

gRv

cr

gRv

22

22

221

VvRr ,when

gRVc

cRgRV

2

2 i.e.

2

22

gRVr

gRv 22 22

2

cr

gRv

cr

gRv

22

22

221

VvRr ,when

gRVc

cRgRV

2

2 i.e.

2

22

gRVr

gRv 22 22

2

If particle never stops then r

cr

gRv

cr

gRv

22

22

221

VvRr ,when

gRVc

cRgRV

2

2 i.e.

2

22

gRVr

gRv 22 22

2

If particle never stops then r

gRV

gRVr

gRvrr

2

22limlim

2

22

2

cr

gRv

cr

gRv

22

22

221

VvRr ,when

gRVc

cRgRV

2

2 i.e.

2

22

gRVr

gRv 22 22

2

If particle never stops then r

gRV

gRVr

gRvrr

2

22limlim

2

22

2

0Now 2 v

cr

gRv

cr

gRv

22

22

221

VvRr ,when

gRVc

cRgRV

2

2 i.e.

2

22

gRVr

gRv 22 22

2

If particle never stops then r

gRV

gRVr

gRvrr

2

22limlim

2

22

2

0Now 2 v022 gRV

cr

gRv

cr

gRv

22

22

221

VvRr ,when

gRVc

cRgRV

2

2 i.e.

2

22

gRVr

gRv 22 22

2

If particle never stops then r

gRV

gRVr

gRvrr

2

22limlim

2

22

2

0Now 2 v022 gRV

gRV 22

cr

gRv

cr

gRv

22

22

221

VvRr ,when

gRVc

cRgRV

2

2 i.e.

2

22

gRVr

gRv 22 22

2

If particle never stops then r

gRV

gRVr

gRvrr

2

22limlim

2

22

2

earth. theescapes particle thei.e.stops,never particle the2 if Thus gRV

0Now 2 v022 gRV

gRV 22

(ii) If ,prove that the time taken to rise to a height R above the earth’s surface is;

gRV 2

gR24

31

(ii) If ,prove that the time taken to rise to a height R above the earth’s surface is;

gRV 2

gR24

31

gRVr

gRv 22 22

2

(ii) If ,prove that the time taken to rise to a height R above the earth’s surface is;

gRV 2

gR24

31

gRVr

gRv 22 22

2

,2 If gRV

(ii) If ,prove that the time taken to rise to a height R above the earth’s surface is;

gRV 2

gR24

31

gRVr

gRv 22 22

2

rgRv

gRgRr

gRv

22

22

2

222

,2 If gRV

(ii) If ,prove that the time taken to rise to a height R above the earth’s surface is;

gRV 2

gR24

31

gRVr

gRv 22 22

2

rgRv

gRgRr

gRv

22

22

2

222

,2 If gRV

rgRv

22 (cannot be –ve, as it does not return)

(ii) If ,prove that the time taken to rise to a height R above the earth’s surface is;

gRV 2

gR24

31

gRVr

gRv 22 22

2

rgRv

gRgRr

gRv

22

22

2

222

,2 If gRV

rgRv

22 (cannot be –ve, as it does not return)

rgR

dtdr 22

(ii) If ,prove that the time taken to rise to a height R above the earth’s surface is;

gRV 2

gR24

31

gRVr

gRv 22 22

2

rgRv

gRgRr

gRv

22

22

2

222

,2 If gRV

rgRv

22 (cannot be –ve, as it does not return)

rgR

dtdr 22

22gRr

drdt

R

R

drrgR

t2

221

R

R

drrgR

t2

221

RRR

R

2 to from g travellinas2

R

R

drrgR

t2

221

RRR

R

2 to from g travellinas2

R

R

rgR

2

23

2 32

21

R

R

drrgR

t2

221

RRR

R

2 to from g travellinas2

R

R

rgR

2

23

2 32

21

RRRRgR

223

22

R

R

drrgR

t2

221

RRR

R

2 to from g travellinas2

R

R

rgR

2

23

2 32

21

RRRRgR

223

22

RRgR

1223

2

R

R

drrgR

t2

221

RRR

R

2 to from g travellinas2

R

R

rgR

2

23

2 32

21

RRRRgR

223

22

RRgR

1223

2

243

g

R

R

R

drrgR

t2

221

RRR

R

2 to from g travellinas2

R

R

rgR

2

23

2 32

21

RRRRgR

223

22

RRgR

1223

2

243

g

R

gR24

31

R

R

drrgR

t2

221

RRR

R

2 to from g travellinas2

R

R

rgR

2

23

2 32

21

RRRRgR

223

22

RRgR

1223

2

243

g

R

gR24

31

seconds 2431 is surface searth'

aboveheight atorisetaken totime

gRR

Exercise 8C; 3, 15, 19