## Forces and Newton’s Laws

The combination of a coordinate system to fix positions and a clock to fix times, constitutes a **Frame of Reference**. When an observer within this frame syas that an object is moving, they are saying the position of the object is changing with time as measured in teh farme of reference that the observer has chosen to use. The way in which a body is described will depend upon the frame of reference from which it is observed.

Newton’s first law is, ‘A body remains at rest or in a state of uniform motion unless it is acted on by an unbalanced force’. An **inertial frame** is a frame of reference in which Newton’s first law holds true. Any frame that moves with constant velocity relative to an inertial frame, while maintaining a fixed orientation, will also be an inertial frame.

Newton’s Second Law: An unbalanced force acting on a body of fixed mass will cause that body to accelerate in the direction of the unbalanced force. The magnitude of the force is equal to the product of the mass and the magnitude of the acceleration, i.e. **F** = m**a**

Mass as used in Newton’s second law, is a measure of the inertia of a body and is often referred to as the **inertial mass** of the body being accelerated. The inertial mass of a body is a measure of its resistance to acceleration.

There are two assumptions about mass that are widely used in Newtonian mechanics. The first is known as the additivity of mass, and asserts that if a body is made up of a number of different parts, the mass of the whole body is the simple sum of its masses. The second in embodied in the law of **Conservation of Mass**. If a body does not exchange matter with its surroundings, its mass remains constant. Einstein’s Special Relativity showed that both these assumptions are wrong, however they are nearly valid over a wide range of conditions.

The unit of force is name after old Isaac, and is called the Newton and 1 N =1 kg m^{-2}

Newton’s Third Law of Motion: If body A exerts a force on body B, then body B exerts a force on body A. These two forces are equal in magnitude, but act in opposite directions.

Associated with each **rigid body** there is a unique point, called the **Centre of Mass**, with the property that if any unbalanced force is applied to the body in such a way that its **line of action** passes through the centre of mass, then the only effect of that force will be to cause **translational acceleration** of the body. If a resultant external force ** F** acts on a body of fixed mass

*m*, then the acceleration of the body’s centre of mass

*a*_{cm}, is given by,

**=**

*F**m*

**a**_{cm}.

All objects near the Earth’s surface are subject to a downward pull due to the Earth’s gravity. In response to this pull, any object that is free to do so will accelerate downwards with and acceleration ** g**, called the

**acceleration due to gravity**.

The Law of Terrestrial Gravity: At any given location on the Earth, all objects that are subject only to the effect of gravity have the same downward acceleration ** g**, irrespective of their mass and composition.

The gravitational pull on the body due to the Earth is called the weight of the body. The weight of a body is a vector and so for a mass *m*, ** W** =

*m*.

**g**

The weight of an object is very different from its mass. Weight is a vector; it is a force, that acts downwards, and is measured in Newtons. Mass is a scalar that quantifies the translational inertia of a body. Mass is an intrinsic property of a body whereas weight arises from a gravitational attraction.

The** Law of Universal Gravitation**: Every particle of matter attracts every other particle of matter with a gravitational force, whose magnitude is directly proportional to the product of the masses of the particles, and inversely proportional to the square of the distance between them.

**F** = [(-Gm_{1}m_{2})/r^{2}]r̂

g = *Gm*_{E}/*r*^{2} where *r* ≥ *R _{E
}*

An upward force * N* is exerted by a rigid surface on which a body is resting. It is at right angles to that surface and is a reaction to the force that the body applies to the surface. For these reasons it is said to be a

**Normal Reaction Force**.

**N**is actually is a result of the impenetrable nature of rigid surfaces. What happens on a real surface is that the weight of the body causes it to sink a little way into the surface, compressing the surface and forcing its atoms closer together, this leads to an increase in the repulsive electrical forces that keep the atoms in the solid apart. The compression of the surface stops when the electrical forces have increased sufficiently to compensate for the force being applied by the body. Thus normal forces are ultimately electrical in nature. The Normal force is an example of a

**contact force**.

When a body exerts a contact force on a rigid surface, the surface exerts a normal reaction on the body. The normal reaction is directed at right angles to the surface and is equal in magnitude to the component of the contact force that is perpendicular to the surface.

Frictional forces resist an applied force until the applied force equals a maximum for the stationary object, *F*_{max} = μ_{static }*N*

Where N is the magnitude of the normal reaction between the surface and the object, and μ_{static} is proportionality constant called the **coefficient of friction**. Note the value of μ does not depend on the area of contact!

The magnitude of the frictional force on a sliding object is given by the force law, ** F** = μ

_{slide }

**where the proportionality constant is called the**

*N***coefficient of sliding friction**. And yes, in general, for two given surfaces, μ

_{slide}< μ

_{static}

Any object that moves through a body of air is subject to a resistive force, called the **aerodynamic drag**, that arises from the flow of air around the moving object. For a sphere of radius *R* moving at a speed *v* through still air; if the sphere is very small and slow moving (*Rv* < 10^{-4}m^{2} s^{-1}), then the force due to air resistance has the magnitude,

*F*_{air} = *k*_{1}*Rv*

Where *k*_{1} = 3.4 x 10-4 N s m^{-2}. If the sphere has moderate size and speed (10^{-4}m^{2} s^{-1}< *Rv* < 1 m^{2} s^{-1}), the force of air resistance has magnitude,

*F*_{air} = *k*_{2}*R*^{2}*v*^{2}

Where *k*_{2} = 0.8 N s^{2} m^{-4}

In practice most everyday objects experience the v^{2} law of air resistance.

When a sphere of radius R moves through a fluid at constant speed v, the magnitude of the viscous force that opposes the motion is,

*F* = 6πη*Rv*

Where η is the **coefficient of viscosity** of the fluid: Typical values for fluids are 1.5 x 10^{-3} N s m^{-2} (water) and 8.4 x 10^{-2} N s m^{-2} (Olive Oil)

If you fix one end of a rope and pull on the other end, the rope will become taut and pull back. The phenomenon that enables the rope to exert this force on you is called tension. It can also arise in string, springs, solid rods, anything that can be stretched. The tension in a body is a measure of its resistance to being extended. Tension is not a force as it has no direction. However it can give rise to **tension forces**.

When the ends of a real body are pulled in opposite directions, the body will generally respond by stretching to some extent. Tension forces tend to restore the body to its unstretched length and are referred to as **restoring forces**. For many bodies, provided they are not stretched too far, the magnitude of the restoring force is directly proportional to the increase in length. Theses bodies are said to obey Hooke’s law: ‘When a body is stretched, the magnitude of the restoring force is directly proportional to the increase in its length, provided the extension is not too great’.

*F*_{x} = –*k*_{s}*x*

Where *k*_{s} is a constant called the spring constant that characterizes the stiffness of the body.

Any spring that obeys Hooke’s law is called an Ideal Spring. Many real springs behave in approximately this way, provided they are not extended or compressed too much. In the case of an ideal spring, a graph showing the magnitude of the restoring force against the extension would be a straight line. For this reason the restoring force produced by such a spring is said to be a **linear restoring force**.

Fluids are incapable of supporting tension, however, fluids can be compressed and they exert forces as a result. These forces act on the walls of any containing vessel, and on any object, even partially immersed in the fluid. They are a consequence of the pressure in the fluid. Like tension, pressure has no particular direction associated with it (it normally acts in all directions), i.e. *P = F/A* where *A* is defined as any area. The unit of pressure is that of a force divided by an area (N/m^{2}) which is usually called a Pascal. 1 Pasacl = 1 N m^{-2
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