X2 T06 01 acceleration (2011)
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Dynamics
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Transcript of X2 T06 01 acceleration (2011)
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
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