# Particle Dynamics

## Frames of Reference

• A frame of reference is a set of axes or coordinates which are used to make measurements of objects inside the frame. For example, a train platform is a frame of reference in which you can measure the motion and orientation of people, trains, etc. Frames of reference are nothing new - it is just a name used to describe where you are measuring from[1].
• An inertial frame of reference is one which is travelling at constant (or zero) velocity relative to another inertial frame. The train station is clearly an inertial frame of reference, since it is not moving (ignoring Earth's rotation). This means that a penny on the tracks which is not experiencing any net force will not be observed to accelerate from the station's frame of reference. A train moving at constant velocity is also an inertial frame of reference, because it is not accelerating. When observed from the train, the penny moves at constant velocity but does not accelerate[2].
• A non-inertial frame of reference is one which is accelerating relative to another frame of reference. If the train in the example above begins to decelerate, it becomes a non-inertial frame of reference[4].

## Newton's Laws

1. If there is zero net force acting on an object, then its acceleration is zero.[5]
1. This is only true in an inertial frame of reference.
2. There may be individual forces acting on an object even if the net force (sum of forces) is zero.
2. F=ma (Force = mass times acceleration)[6]
3. For every action, there is an equal and opposite reaction.[7]
1. This means that forces occur in pairs acting in opposite directions.
2. If two objects A and B interact, the force on A due to B is equal and in the opposite direction to the force on B due to A.
3. Since the forces are equal and opposite, only external forces will affect the acceleration of the system. The internal reaction forces will cancel each other out. So if an inelastic string attached to a cart is pulled by an external force, the forces between the cart and the string will cancel out and the net force on the cart will be the same as that on the string.

## Mass

Mass is defined as an object's resistance to changes in velocity.[8] For objects with more mass, a greater force is needed to cause the same change in velocity (the same acceleration). This relationship is expressed in Newton's second law.

### Weight

Weight (W) is a measurement of the downwards force exerted by an object due to its mass (m) and gravitational acceleration (g).[9]

W = -mg

Weight is a force, and so is measured in Newtons (N). The above formula is an application of Newton's second law.

It is important to note that while weight is affected by gravity, mass is not. On a planet with a stronger gravitational field than earth, an object's weight will be greater than on earth because the gravitational acceleration is greater. The object's mass does not change.

### "Light" objects

In physics, the term "light object" is used to describe an object which has negligible mass (such as the string pulling a cart). This mean's the object's mass does not need to be considered when performing calculations.[10]

## Friction and Contact Forces

When an object is in contact with a surface, it will experience two forces.[11]

• Normal force is directed at a normal to the surface.[12]
• Friction force is directed tangential to the surface, in the direction which opposes motion.[13]

The vector sum of these two forces is the contact force.

### Static Friction

Static Friction (Fs) exists between two touching surfaces that are not moving relative to one another (not slipping). Static friction is proportional to the normal force (N) and the coefficient of static friction (μs). [14]

Fs ≤ μsN

• Static friction opposes forces which will cause an object to slide along a surface. It will exactly balance out any motion-causing force up to the maximum value of μsN. The inequality sign in the equation indicates that the friction force does not have a constant value.
• Friction will only resist forces (or force components)tangential to the surface.
• When the friction force reaches its maximum value (μsN), the situation is called impending motion. If the motion-causing force increases beyond this, the object will begin to move and static friction will cease to apply.
• The coefficient of static friction is a property of the surfaces involved, and generally has values from 0.3-1.0. Substances which cause greater friction (eg. rubber) have higher values.

Be careful: The force that causes motion is not the normal force. The normal force is only the net force component acting perpendicular to the surface.

### Kinetic Friction

If two contacting surfaces are moving relative to one another, kinetic friction exists between them. Kinetic friction (Fk) is proportional to the normal force (N) and the coefficient of kinetic friction (μs).[15]

Fk = μkN

• Unlike static friction, kinetic friction is constant (for a constant normal force).
• The coefficient of kinetic friction is generally less than that of static friction. This is one reason why a larger force is needed to start an object moving than to keep it going.

## Centripetal Acceleration and Force

An object undergoing circular motion experiences centripetal acceleration. This acceleration is provided by a centripetal force acting towards the center of the circle. The centripetal force (Fc) on an object may be calculated from the object's mass (m) and centripetal acceleration (ac) using Newton's second law[16].

Fc = mac

Using the formula for centripetal acceleration, this may be re-written as:

Fc = m(v2/r)

A common application of centripetal acceleration is gravity holding objects in orbit. In this case, the gravitational force and centripetal force will be the same.

## End

This is the end of this topic. Click here to go back to the main subject page for Higher Physics 1A.

## References

Textbook refers to Serway & Jewett, Physics for Scientists and Engineers (Brooks/Cole , 8th ed, 2010)
(Slides) refers to those distributed by Wolfe, J (2012) on his First Year Physics site

1. Textbook, pp.105-106
2. Textbook, p. 106
3. Textbook, p. 106
4. Textbook, p. 106
5. (Slides) Particle dynamics, p. 1
6. (Slides) Particle dynamics, p. 2
7. (Slides) Particle dynamics, pp. 3-6
8. Textbook, pp.106-107
9. Textbook, pp.109-110
10. (Slides) Particle Dynamics, p. 7
11. (Slides) Particle Dynamics, p. 14
12. Textbook, p. 111
13. Textbook, p122
14. Textbook, pp. 122-124
15. Textbook, pp. 122-124
16. Textbook, pp. 139-140