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