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# Acceleration of Gravity and Newton's Second Law

Acceleration of Gravity is one of the most used physical constants - known from

### Newton's Second Law

"Change of motion is proportional to the force applied, and take place along the straight line the force acts."

Newton's second law for the gravity force - weight - can be expressed as

W = F g

= m a g

= m g                                 (1)

where

W, F g = weight, gravity force (N, lb f )

m = mass (kg, slugs )

a g = g = acceleration of gravity (9.81 m/s 2 , 32.17405 ft/s 2 )

The force caused by gravity - a g - is called weight .

Note!

• mass is a property - a quantity with magnitude
• force is a vector - a quantity with magnitude and direction

The acceleration of gravity can be observed by measuring the change of velocity related to change of time for a free falling object:

a g = dv / dt (2)

where

dv = change in velocity (m/s, ft/s)

dt = change in time (s)

An object dropped in free air accelerates to speed 9.81 m/s (32.174 ft/s) in one - 1 - second .

• a heavy and a light body near the earth will fall to the earth with the same acceleration (when neglecting the air resistance)

### Acceleration of Gravity in SI Units

1 a g = 1 g = 9.81 m/s 2 = 35.30394 (km/h)/s

### Acceleration of Gravity in Imperial Units

1 a g = 1 g = 32.174 ft/s 2 = 386.1 in/s 2 = 22 mph/s

### Velocity and Distance Traveled by a Free Falling Object

The velocity for a free falling object after some time can be calculated as:

v = a g t (3)

where

v = velocity (m/s)

The distance traveled by a free falling object after some time can be expressed as:

s = 1/2 a g t 2 (4)

where

s = distance traveled by the object (m)

The velocity and distance traveled by a free falling object:

Free Falling Object - Velocity and Distance Traveled vs. Time
Time
(s)
VelocityDistance
m/s km/h ft/s mph m ft
1 9.8 35.3 32.2 21.9 4.9 16.1
2 19.6 70.6 64.3 43.8 19.6 64.3
3 29.4 106 96.5 65.8 44.1 144.8
4 39.2 141 128.7 87.7 78.5 257.4
5 49.1 177 160.9 110 122.6 402.2
6 58.9 212 193.0 132 176.6 579.1
7 68.7 247 225.2 154 240.3 788.3
8 78.5 283 257.4 176 313.9 1,029.6
9 88.3 318 289.6 198 397.3 1,303.0
10 98.1 353 321.7 219 490.5 1,608.7

Note! Velocities and distances are achieved without aerodynamic resistance ( vacuum conditions). The air resistance - or drag force - for objects at higher velocities can be significant - depending on shape and surface area.

#### Example - Free Falling Stone

A stone is dropped from 1470 ft (448 m) - approximately the height of Empire State Building. The time it takes to reach the ground (without air resistance) can be calculated by rearranging (4) :

t = (2 s / a g ) 1/2

= (2 (1470 ft) / (32.174 ft/s 2 )) 1/2

= 9.6 s

The velocity of the stone when it hits the ground can be calculated with (3) :

v = (32.174 ft/s 2 ) (9.6 s)

= 308 ft/s

= 210 mph

= 94 m/s

= 338 km/h

#### Example - A Ball Thrown Straight Up

A ball is thrown straight up with an initial velocity of 25 m/s . The time before the ball stops and start falling down can be calculated by modifying (3) to

t = v / a g

= (25 m/s) / (9.81 m/s 2 )

= 2.55 s

The distance traveled by the ball before it turns and start falling down can be calculated by using (4) as

s = 1/2 (9.81 m/s 2 ) ( 2.55 s ) 2

= 31.8 m

### Newton's First Law

"Every body continues in a state of rest or in a uniform motion in a straight line, until it is compelled by a force to change its state of rest or motion."

### Newton's Third Law

"To every action there is always an equal reaction - if a force acts to change the state of motion of a body, the body offers a resistance equal and directly opposite to the force."

### Common Expressions

• mass loads: kg/m 2 or kg/m 3
• stress: N/mm 2
• bending moment: kNm
• shear: kN
• 1 N/mm = 1 kN/m
• 1 N/mm 2 = 10 3 kN/m 2
• 1 kNm = 10 6 Nmm

### Latitude and Acceleration of Gravity

Acceleration of gravity varies with latitude - examples:

Acceleration of Gravity vs. Location and Latitude
LocationLatitudeAcceleration og Gravity
(m/s 2 )
North Pole 90° 0' 9.8321
Anchorage 61° 10' 9.8218
Greenwich 51° 29' 9.8119
Paris 48° 50' 9.8094
Washington 38° 53' 9.8011
Panama 8° 55' 9.7822
Equator 0° 0' 9.7799

## Related Topics

### • Basics

The SI-system, unit converters, physical constants, drawing scales and more.

### • Dynamics

Motion - velocity and acceleration, forces and torque.

### • Mechanics

Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

## Related Documents

### Acceleration

Change in velocity vs. time used.

### Acceleration of Gravity vs. Latitude and Elevation

Acceleration of gravity due to latitude and elevation above sea level.

### Acceleration Units Converter

Converting between units of acceleration.

### Banked Turn

A turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn.

### Bodies Moving on Inclined Planes - Acting Forces

Required forces to move bodies up inclined planes.

### Car Acceleration

Car acceleration calculator.

### Center of Gravity

A body and the center of gravity.

### Center of Gravity and Buoyancy

Stability - the center of gravity vs. the center of buoyancy.

### Centripetal and Centrifugal Acceleration Force

Forces due to circular motion and centripetal / centrifugal acceleration.

### Density vs. Specific Weight and Specific Gravity

An introduction to density, specific weight and specific gravity.

### Elevators - Force and Power

Required force and power to lift an elevator.

### Force

Newton's third law - force vs. mass and acceleration.

### Force Ratio

The force ratio is the load force versus the effort force.

### Forces acting on Body Moving in a Horizontal Plane

The forces acting on bodies moved in horizontal planes.

### Mass vs. Weight

Mass vs. weight - the Gravity Force.

### Projectile Range

Calculate the range of a projectile - a motion in two dimensions.

### SI System

An introduction to the SI metric system.

### Solar System Constants

Properties and data for the Sun, the Earth and the Moon.

### Support Reactions - Equilibrium

Static equilibrium is achieved when the resultant force and resultant moment equals to zero.

### Toggle Joint

A toggle joint mechanism can be used to multiply force.

### Universal Gravitational Law

Gravitational attraction between two objects vs. mass of the objects and the distance between them.

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