# Acceleration of Gravity and Newton's Second Law

## Acceleration of gravity and Newton's Second Law - SI and Imperial units

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, Fg= 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/s22 mph/s^{2 }=^{}

### 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 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 |