# Physics equations/Impulse, momentum, and motion about a fixed axis

## Contents

##### Impulse and momentum[edit]

**Impulse** is denoted by the symbols *I* (which is too easily confused with moment of inertia), or *Imp*, or **J** (we shall use the latter):^{[1]}

**Momentum** (or *linear momentum*) is:

Total linear momentum is an **extrinsic** and *conserved* quantity, provided the net external force is zero. It **can be shown** that momentum obeys *(d/dt)Σ p=ΣF_{ext}*. Kinetic energy and momentum are related by,K=½mv

^{2}=p

^{2}/(2m). Linear momentum is related to linear momentum by the

**impulse-momentum theorem**:

From Newton's second law, force is related to momentum **p** by

- . Therefore,

**p**is the change in linear momentum from time

*t*

_{1}to

*t*

_{2}. This is often called the impulse-momentum theorem.

#### Torque[edit]

**Torque**, **moment** or **moment of force** is also called moment. The symbol for torque is typically *τ*, the Greek letter *tau*. When it is called moment, it is commonly denoted *M*.^{[2]} The SI units for torque is the newton metre (N·m). It would be inadvisable to call this a Joule, even though a Joule is also a (N·m).

More generally, the torque on a particle (which has the position **r** in some reference frame) can be defined as the cross product:

where **r** is the particle's position vector relative to the fulcrum, and **F** is the force acting on the particle. The magnitude *τ* of the torque is given by

where *r* is the distance from the axis of rotation to the particle, *F* is the magnitude of the force applied, and *θ* is the angle between the position and force vectors. Alternatively,

where *F*_{⊥} is the amount of force directed perpendicularly to the position of the particle and *r*_{⊥} is called the lever arm.

#### Rotational motion about a fixed axis[edit]

Angular displacement may be measured in radians or degrees. If using radians, it provides a very simple relationship between distance traveled around the circle and the distance *r* from the center. ^{[3]}

A particle moves in a circle of radius . Having moved an arc length , its angular position is relative to its original position, where .

In mathematics and physics it is usual to use the natural unit radians rather than degrees or revolutions. Units are converted as follows:

An angular displacement is described as

#### Angular speed and angular velocity[edit]

Angular velocity is the change in angular displacement per unit time. The symbol for angular velocity is and the units are typically rad s^{−1}. Angular speed is the magnitude of angular velocity.

The instantaneous angular velocity is related to particles speed by

where is the transitional speed of the particle. A changing angular velocity indicates the presence of an angular acceleration in rigid body, typically measured in rad s^{−2}. The average angular acceleration over a time interval Δ*t* is given by

The instantaneous acceleration *α*(*t*) is given by

#### Kinematic equations of motion[edit]

When the angular acceleration is constant, the five quantities angular displacement , initial angular velocity , final angular velocity , angular acceleration , and time can be related by four equations of kinematics:

#### Kinetic energy[edit]

The kinetic energy of a rigid system of particles moving in the plane is given by^{[4]}

Thus, where is called the **moment of inertia**.

The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, with the summation replaced by the integral,

Here **r** is the distance to the axis and ρ=ρ(**r**) is the mass density. As shown above, this can be converted into a line, surface, or volume integral for a substance with a surface mass density ρ(x,y) or line mass density λ(x)

#### Torque, angular momentum, and work[edit]

The rotational equivalent of Newton's ,*F = ma*, linear momentum as, *p = mv*, and work as *W = FΔx → ʃFdx*, is , , and , respectively.Here, *L* is angular momentum, which does not have the same units as linear momentum. But work, *W*, is measured in the same units (Joules).

#### [edit]

Description^{[5]} |
Figure | Moment(s) of inertia |
---|---|---|

Point mass m at a distance r from the axis of rotation. |
||

Two point masses, M and m, with reduced mass and separated by a distance, x. |
||

Rod of length L and mass m(Axis of rotation at the end of the rod) |
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Rod of length L and mass m |
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Thin circular hoop of radius r and mass m |
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Thin cylindrical shell with open ends, of radius r and mass m |
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Solid cylinder of radius r, height h and mass m |
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Sphere (hollow) of radius r and mass m |
||

Ball (solid) of radius r and mass m |
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Thin rectangular plate of height h and of width w and mass m(Axis of rotation at the end of the plate) |
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Solid cuboid of height h, width w, and depth d, and mass m |

- ↑ //en.wikipedia.org/w/index.php?title=Impulse_(physics)&oldid=580814154
- ↑ https://en.wikipedia.org/w/index.php?title=Torque&oldid=582917749
- ↑ https://en.wikipedia.org/w/index.php?title=Angular_displacement&oldid=575169747
- ↑ https://en.wikipedia.org/w/index.php?title=Moment_of_inertia&oldid=583286938
- ↑ https://en.wikipedia.org/w/index.php?title=List_of_moments_of_inertia&oldid=582953751