Section 8.5 Applications to Physics. In physics the word “work” is used to describe the work a...
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Transcript of Section 8.5 Applications to Physics. In physics the word “work” is used to describe the work a...
Section 85Applications to Physics
bull In physics the word ldquoworkrdquo is used to describe the work a force has done on an object to move it some distancendash Work done = Force Distance or W = F D
Units
Force Distance Work
International Units (SI)
Newton (nt) Meter (m) Joule (j)
British Units Pound (lb) Foot (ft) Foot-pound (ft-lb)
bull If an object of mass m moves along a straight line given by s(t) then the force (in the same direction) is defined by
bull What is the work required to raise a 5 kg mass up 10 meters
2
2
dt
sdmF
What if the force is not constantbull Consider a force that varies along a to b
ndash Call if f(x)
bull Divide the interval a to b into n subintervals
bull Pick in the ith interval
bull Then is the force
bull The interval is then small enough so that the force is constant
bull Then
bull So
ix
)( ixf
n
iiii xxfwthenxxfw
1
)()(
b
adxxfw )(
Hookersquos Law
bull The force required to maintain a spring stretched x units beyond its natural length is proportional to x (let k be the constant of proportionality) so we getndash F = kx
Examplebull A spring has a natural length of 20 cm If a 25
newton force is required to keep it stretched to 30 cm how much work is required to stretch it from 20 cm to 25 cm
Examplebull A trough that has a triangular cross section that
is 5m high 3m wide at the top and 8m long is filled up to 3 meters with water Given that the density of water is 1000 kgm3 how much work is required in order to empty the trough
Force and Pressurebull Can use a definite integral to compute the
force exerted by a liquid on a surface
bull The force is directly related to the pressurendash Pressure of a liquid is the force per unit area
exerted by the liquidndash It is equal in all directionsndash It increases with depth
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
bull In physics the word ldquoworkrdquo is used to describe the work a force has done on an object to move it some distancendash Work done = Force Distance or W = F D
Units
Force Distance Work
International Units (SI)
Newton (nt) Meter (m) Joule (j)
British Units Pound (lb) Foot (ft) Foot-pound (ft-lb)
bull If an object of mass m moves along a straight line given by s(t) then the force (in the same direction) is defined by
bull What is the work required to raise a 5 kg mass up 10 meters
2
2
dt
sdmF
What if the force is not constantbull Consider a force that varies along a to b
ndash Call if f(x)
bull Divide the interval a to b into n subintervals
bull Pick in the ith interval
bull Then is the force
bull The interval is then small enough so that the force is constant
bull Then
bull So
ix
)( ixf
n
iiii xxfwthenxxfw
1
)()(
b
adxxfw )(
Hookersquos Law
bull The force required to maintain a spring stretched x units beyond its natural length is proportional to x (let k be the constant of proportionality) so we getndash F = kx
Examplebull A spring has a natural length of 20 cm If a 25
newton force is required to keep it stretched to 30 cm how much work is required to stretch it from 20 cm to 25 cm
Examplebull A trough that has a triangular cross section that
is 5m high 3m wide at the top and 8m long is filled up to 3 meters with water Given that the density of water is 1000 kgm3 how much work is required in order to empty the trough
Force and Pressurebull Can use a definite integral to compute the
force exerted by a liquid on a surface
bull The force is directly related to the pressurendash Pressure of a liquid is the force per unit area
exerted by the liquidndash It is equal in all directionsndash It increases with depth
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
bull If an object of mass m moves along a straight line given by s(t) then the force (in the same direction) is defined by
bull What is the work required to raise a 5 kg mass up 10 meters
2
2
dt
sdmF
What if the force is not constantbull Consider a force that varies along a to b
ndash Call if f(x)
bull Divide the interval a to b into n subintervals
bull Pick in the ith interval
bull Then is the force
bull The interval is then small enough so that the force is constant
bull Then
bull So
ix
)( ixf
n
iiii xxfwthenxxfw
1
)()(
b
adxxfw )(
Hookersquos Law
bull The force required to maintain a spring stretched x units beyond its natural length is proportional to x (let k be the constant of proportionality) so we getndash F = kx
Examplebull A spring has a natural length of 20 cm If a 25
newton force is required to keep it stretched to 30 cm how much work is required to stretch it from 20 cm to 25 cm
Examplebull A trough that has a triangular cross section that
is 5m high 3m wide at the top and 8m long is filled up to 3 meters with water Given that the density of water is 1000 kgm3 how much work is required in order to empty the trough
Force and Pressurebull Can use a definite integral to compute the
force exerted by a liquid on a surface
bull The force is directly related to the pressurendash Pressure of a liquid is the force per unit area
exerted by the liquidndash It is equal in all directionsndash It increases with depth
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
What if the force is not constantbull Consider a force that varies along a to b
ndash Call if f(x)
bull Divide the interval a to b into n subintervals
bull Pick in the ith interval
bull Then is the force
bull The interval is then small enough so that the force is constant
bull Then
bull So
ix
)( ixf
n
iiii xxfwthenxxfw
1
)()(
b
adxxfw )(
Hookersquos Law
bull The force required to maintain a spring stretched x units beyond its natural length is proportional to x (let k be the constant of proportionality) so we getndash F = kx
Examplebull A spring has a natural length of 20 cm If a 25
newton force is required to keep it stretched to 30 cm how much work is required to stretch it from 20 cm to 25 cm
Examplebull A trough that has a triangular cross section that
is 5m high 3m wide at the top and 8m long is filled up to 3 meters with water Given that the density of water is 1000 kgm3 how much work is required in order to empty the trough
Force and Pressurebull Can use a definite integral to compute the
force exerted by a liquid on a surface
bull The force is directly related to the pressurendash Pressure of a liquid is the force per unit area
exerted by the liquidndash It is equal in all directionsndash It increases with depth
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
Hookersquos Law
bull The force required to maintain a spring stretched x units beyond its natural length is proportional to x (let k be the constant of proportionality) so we getndash F = kx
Examplebull A spring has a natural length of 20 cm If a 25
newton force is required to keep it stretched to 30 cm how much work is required to stretch it from 20 cm to 25 cm
Examplebull A trough that has a triangular cross section that
is 5m high 3m wide at the top and 8m long is filled up to 3 meters with water Given that the density of water is 1000 kgm3 how much work is required in order to empty the trough
Force and Pressurebull Can use a definite integral to compute the
force exerted by a liquid on a surface
bull The force is directly related to the pressurendash Pressure of a liquid is the force per unit area
exerted by the liquidndash It is equal in all directionsndash It increases with depth
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
Examplebull A spring has a natural length of 20 cm If a 25
newton force is required to keep it stretched to 30 cm how much work is required to stretch it from 20 cm to 25 cm
Examplebull A trough that has a triangular cross section that
is 5m high 3m wide at the top and 8m long is filled up to 3 meters with water Given that the density of water is 1000 kgm3 how much work is required in order to empty the trough
Force and Pressurebull Can use a definite integral to compute the
force exerted by a liquid on a surface
bull The force is directly related to the pressurendash Pressure of a liquid is the force per unit area
exerted by the liquidndash It is equal in all directionsndash It increases with depth
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
Examplebull A trough that has a triangular cross section that
is 5m high 3m wide at the top and 8m long is filled up to 3 meters with water Given that the density of water is 1000 kgm3 how much work is required in order to empty the trough
Force and Pressurebull Can use a definite integral to compute the
force exerted by a liquid on a surface
bull The force is directly related to the pressurendash Pressure of a liquid is the force per unit area
exerted by the liquidndash It is equal in all directionsndash It increases with depth
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
Force and Pressurebull Can use a definite integral to compute the
force exerted by a liquid on a surface
bull The force is directly related to the pressurendash Pressure of a liquid is the force per unit area
exerted by the liquidndash It is equal in all directionsndash It increases with depth
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
bull At a depth of h meters the pressure p exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter
bull If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity The weight of the column is δgh so Pressure = Mass density g Depth or P = δgh
bull Provided the pressure is constant over that area we have Force = Pressure Area
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
Units
Force Area Pressure
International Units (SI)
Newton (nt) Meter2 (m2) Ntm2 called pascal
(mass)
British Units Pound (lb) Foot2 (ft2) Lbft2
(weight)
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-
Examplebull 24 The Three Gorges Dam is currently being
built in China When it is finished in 2009 it will be the largest damn in the world about 2000 m long and 180 m high creating a lake the length of Lake Superior Assume the damn is rectangular in shape
1 Estimate the water pressure at the base of the dam
2 Set up and evaluate a definite integral giving the total force of the water on the dam
- Section 85 Applications to Physics
- Slide 2
- Slide 3
- What if the force is not constant
- Hookersquos Law
- Example
- Slide 7
- Force and Pressure
- Slide 9
- Units
- Slide 11
-