The Physics of Skyscrapers

Potential and Kinetic Energy

Diagram

Tuned Mass Damper

Animation

Newton's 3rd Law

Video of Tuned Mass Damper

Newton's Laws of Motion

Newton's 3rd Law

How does a Skyscraper Resist Wind?

PE=m x a x h

PE=(.0025 kg)(9.8m/s^2)(1609m)

PE=39.42 J

PE = 39.42 J

Fa = -Fb

Force of Wind = -Force of Tuned Mass Damper

A 100mph gust of wind exerts a force of 3,000 newtons on the top of a 1,500ft building. What force does the tuned mass damper exert back?

PE near top = 10 calories

Fa=-Fb

6,000N=-6,000N

-6,000N

Fa=Fb

9,000N=-9,000N

-9,000N

1: Object at rest stays at rest; object in motion stays in motion

2: Acceleration is directly related to net force but indirectly related to mass (or, F=ma)

For every action, there is an equal and opposite reaction.

Force of A = -Force of B

Direct Relationship

Fa = -Fb

Force = mass (kg) x acceleration (m/s^2)

Potential and Kinetic Energy

[mass is the measure of an object's matter]

[acceleration is change of velocity per unit of time]

3: For every action, there is an equal and opposite reaction

Force of Wind = - Force of TMD

Direct Relationship

Taipei 101

What if you dropped the penny from the top of a future mile high (1609 m) skyscraper? What is its potential energy at the top?

Diagram

All PE

Newton's 3rd Law

Potential

to

Kinetic

Fa=-Fb

3,000N=-3,000N

-3,000N

Force of TMD = -3,000 N

All KE

This force is about 5 times the force that the earth’s gravity exerts on the average human.

What if the wind’s force doubles, to 6,000 N? Triples to 9,000 N?

Potential and Kinetic Energy

PE=m x a x h

PE=.0025kg x 9.8m/s^2 x 442m

PE=10.83 J

Direct Relationship

PE = 10.83 J

KE = 2.53 J

KE=1/2mv^2

KE=(1/2)(.0025kg)(45m/s)^2

KE=2.53 J

Potential and Kinetic Energy

PE near top = 2.5 calories

KE near bottom = 0.5 calories

Terminal velocity is the reason for this difference.

Preview

World's Tallest Skyscrapers

If you were to drop a 2.5g penny from the top of the 442 meter Sears Tower, what would be its potential energy before you dropped it? What would be its kinetic energy 2 meters from the ground? Assume its terminal velocity is 45 m/s.

PE = m x a x h

KE = 1/2mv^2

KE = kinetic energy (J)

m = mass (kg)

[mass is the measure of an object's amount of matter]

PE = potential energy (J)

m = mass (kg)

[mass is the measure of an object's amount of matter]

a = acceleration (m/s^2)

[acceleration is the rate at which an object changes

its velocity]

h = height (m)

[height is the distance from the place of zero PE]

PE=m x a x h

KE=1/2mv^2

v = velocity (m/s)

[velocity is the amount of distance covered per unit of time]

1. Wind Resistance

2. Gravitational Effects

3. Force of Earthquakes

4. Energy of a Dropped Object

Potential and Kinetic Energy

Video

Potential and Kinetic Energy

Hong Kong: 6,439 Skyscrapers

Distance vs. Gravitational Force

"If you dropped a penny from the Empire State Building, it would kill a person on impact."

As you get further from Earth's surface, the gravity you experience decreases.

Inverse Relationship

Fg = 1/d^2

Potential and Kinetic Energy

Is there enough

kinetic energy?

Potential Energy results from the position of an object.

Kinetic Energy results from the motion of an object.

Review

The Physics of Skyscrapers

Distance vs. Gravitational Force

Home Insurance Building, Chicago

1885

PE=m x a x h Greater the height, greater the PE.

KE=1/2mv^2 Greater the velocity, greater the KE.

1. Wind Resistance

2. Gravitational Effects

3. Force of Earthquakes

4. Energy of a Dropped Object

Because distance is inversely related to gravity,

could a skyscraper that is built tall enough cause

a noticeable difference in the force of gravity?

Could it be tall enough to cause a person on the

top floor to age faster than a person on the bottom

floor?

How does gravity feel on the top floor of the world's tallest building?

What does the future hold?

My kind of town...

Distance vs. Gravitational Force

Demo

Fg=G(m1m2)/d^2

Fg = force of gravity exerted on each body (N)

G=6.67 x 10^-11

[gravitational constant]

m1 & m2 = mass of two bodies (kg)

[mass is the measure of an object's matter]

Force on small building

vs. Force on tall building

d = distance between the two bodies (m)

[distance is displacement]

What is the force of gravity on a 70kg person staying on the 163rd floor of the Burj Dubai (about 830 meters)? What about the force on the same person if he or she is on the 1st floor (about 1 meter)?

The Burj Dubai weighs 498,961,607 kg, and the distance from earth's surface to its center is 6,371,000 m.

Fg=G(m1m2)/d^2

G=6.67 x 10^-11

Direct Relationship

Diagram

San Francisco Earthquake, 1906

Newton's Laws of Motion

Newton's 2nd Law

Distance vs. Gravitational Force

6,371,830 m from Earth's center

Newton's 2nd Law

1: Object at rest stays at rest; object in motion stays in motion

F=ma

F=(136,077,711kg)(0.5 m/s^2)

F=68,038,855.5 N

Top Floor:

Fg=G(m1m2)/d^2

Fg=6.67x10^-11(70kgx(6x10^24kg))/(6,378,047^2)

Fg=688.652N

How much force would be exerted on SF's Transamerica Building in an earthquake? The building is roughly 136,077,711 kg, and during a severe earthquake, the ground can accelerate at roughly 0.5 m/s^2.

F=ma

F=(201,849,000kg)(0.5m/s^2)

F=100,924,500 N

This is more than 30,000,000 more Newtons of force than the San Francisco tower experienced. Thankfully we aren’t as likely to have earthquakes in Chicago.

The bigger the mass, the bigger the force. The larger

the acceleration, the larger the force.

F = ma

Direct Relationship

Fg = 688.652 N vs Fg = 690.176N

2: Acceleration is directly related to Force but indirectly related to mass (or, F=ma)

F = 68,038,855.5 N

The distance between the first floor and the

top floor is about 1.5 Newtons.

This difference is the equivalent to

about 1.5 apples.

F= ma

6,371,001 m from Earth's center

This is the same force as a 1 ton car hitting you

straight on at 168 miles per hour. (75,000 m/s).

This shows the importance of engineering a

building to withstand even the smallest vibrations.

First Floor:

Fg=G(m1m2)/d^2

Fg=6.67x10^-11(70kgx(6x10^24kg))/(6,371,001^2)

Fg=690.176 N

F = force (N)

m = mass (kg)

[mass is the measure of an object's amount of matter]

a = acceleration (m/s^2)

[acceleration is the rate at which an object changes its velocity]

3: For every action there is an equal and opposite reaction

Earthquake Force Video

Distance vs. Gravitational Force

Diagram

Distance vs. Gravitational Force

How could San Francisco withstand an earthquake today?

Newton's 2nd Law

136,077,711kg

What if a 5 mile (8047 meter) high skyscraper was built? What would be the force of gravity on a 70 kg person on the top floor opposed to the first floor (1 meter from surface)?

163rd floor:

Fg=G(m1m2)/d^2

Fg=6.67x10^-11(70kgx(6x10^24kg))/(6,371,830^2)

Fg=689.996N

What if an earthquake of the same strength stuck Chicago and the 201,849,000 kg Sears Tower? How much force would be exerted on the Sears Tower?

1st floor:Fg=G(m1m2)/d^2

Fg=6.67x10^-11(70kgx(6x10^24kg))/(6,371,001^2)

Fg=690.176 N

Acceleration

0.5 m/s^2

Fg = 689.996 N vs Fg = 690.176 N

Transamerica Pyramid

Gravitational force on first floor is 0.18 N greater than on the top floor.

0.18 N is roughly equal to 1/5 of an apple's weight.

The person on the first floor will age slower than the person at the top, but only by a minuscule amount.

Inverse Relationship