# 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:

Time (s) | Velocity | Distance | ||||
---|---|---|---|---|---|---|

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

- superimposed loads:
*kN/m*^{2} - mass loads:
*kg/m*or^{2}*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:

Location | Latitude | Acceleration of 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

Basic engineering data. SI-system, unit converters, physical constants, drawing scales and more.

### • Dynamics

Motion of bodies and the action of forces in producing or changing their motion - velocity and acceleration, forces and torque.

### • Mechanics

The relationships between forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

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