The forces acting on the spheres are shown below.
The forces of gravitational attraction between the gold masses and the lead masses causes the bar to twist through an angle
The torque on the beam is given byand equating this to the restoring torque whereis the restoring force per radian of turn exerted by the torsion bar givesthen
]]>The force may be a tension in a string, gravitational, the force between electrically charged particles, a reaction force or have another origin. The magnitude of this centripetal force iswhere
is the mass of the body performing circular motion
is it's speed
is the radius of the motion
The centrifugal force also has magnitudebut is directed outwards as shown above.
]]>Example: The force of gravity which keeps the Earth moving around the Sum, with an orbit very nearly a circle.
Example: The force of friction between the road and a car's tyres keeps the car on the road when the car rounds a bend.
Example: The tension in a string hanging vertically, attached to a particle moving in a horizontal circle.]]>
All bodies in motion have angular momentum about some point, even if the the particle is travelling in a straight line. As the particle moves from A to B above the quantitystays constant. Angular momentum may be more usefully visualised the particles moving in a circle.
For a particle moving in a circleandare at right angles, andare at right angles to bothandThis means that
Kepler's Laws of Motion are a direct consequence of the Law of Conservation of Angular Momentum. The Law applies on all scales, including the quantum mechanical scale, and is responsible for the stability of atoms.
]]>It is used to overcome air resistance.
Some is used to overcome friction between the road and the wheels, in the internal machinery of the car.
The air in the wheels is set in motion and experiences heating.
Some is used to increase the kinetic energy of the car, or changed to heat in the brakes when the car slows down.
Some is used to overcome the force of gravity if the car moves up the slope, or, if the car is moving down the slope, the gravitational potential energy lost is changed into other forms of energy.
In general we can say that the power provided by the engine is equal to the sum of the powers produced in each of the processes described above. We consider only friction, kinetic energy and gravitational potential energy here. A car is shown below moving up a hill at an angleto the horizontal.
The driving force supplied by the cars engine isInseconds the car moves a distanceso the engine supplies energy equal to
The resistance to motion isso the energy needed to overcome it is, by the same argument as above, equal toAs shown by the diagram below, the height of the car increases byin 1 second, so the increase in gravitational potential energy inseconds is
Finally, suppose that the velocity of the car changes fromtoinseconds, The increase in kinetic energy is
Putting all these together gives
Dividing bygives
Now letthenis the instantaneous acceleration andso and
Finally, in this expression,is a common factor so
]]>In the same way, a glider moving on a frictionless track hitting a spring has kinetic energy changing into potential energy stored in the spring. The spring is compressed. Eventually the kinetic energy is reduced to zero, and when the spring starts to expand again, potential energy is changed back into kinetic energy. If there is no friction either between glider and track or internal energy in the spring, Then all the kinetic energy is recovered.
In both these cases the sum of kinetic energy plus potential energy is conserved. A force that offers this sort of energ store to enable the exchange of potential and kinstic energy is called a conservative force.
The essential features of conservative forces are:
A body may move between two points by various paths but the work done in moving from point to point is independent of the path taken:below.
The potential energy of a particle depends only on it's position and not on it's velocity or acceleration.
When the start and end points are the same the total work done is zero.
The work done by conservative forces can always be expressed as the difference between the initial and final values of a potential energy function.
Friction and air resistance are not conservative forces. Work must be done to overcome friction and air resistance and potential energy does not increase as a result of overcoming friction and air resistance. Instead, in these cases, heat is generated, which cannot be later used to do work.
A contact force has two components. The part of the force that lies within the plane of contact is friction, which must be overcome for the two objects to slide relative to one another along that plane. The part of the force that is perpendicular to the plane of contact is called the normal force. The maximum possible frictional forces,is proportional to the normal contact forceand the constant of proportionality is denoted byThusIfand there is no movement then it is called limiting friction. If the two objects are in relative motion then
Strictly speaking, contact forces are only a useful simplification for use in classical mechanics. Everyday objects on Earth do not actually touch each other; rather contact forces are the result of the interactions of the electrons at or near the surfaces of the objects (exchange interaction).
]]>The moment of a forceis only defined with respect to a certain point P – the moment of about P - and in general when P is changed, the moment changes. However, the moment (torque) of a couple is independent of the reference point P : Any point will give the same moment. In other words, a torque vector, unlike any other moment vector, is a "free vector".
Proof: Suppose there are a set of force vectorsetc. that form a couple, with position vectors (about some origin P )etc., respectively. The moment about P is
Now we pick a new reference point P' that differs from P by the vector r . The new moment is
We can simplify this as follows:
Therefore,
This proves that the moment is independent of reference point, which is proof that a couple is a free vector.
]]>To move from a distancesay, near to the Earth's surface to a distancewe need to put in an amount of energy per unit mass for the object being moved of
since
the acceleration due due gravity is given by the force of gravity on a 1 Kg mass at the Earth's surface.
The increase in Gravitational Potential is thenand the increase in gravitational potential energy for a massis]]>
This means that a man standing on the equator is experiencing a centrifugal force equal toas a result of his motion around the centre of the Earth (the speed of the man on the equator around the centre of the Earth is ). Whether the man is on the equator or at the pole, assuming the distance towards the centre of the Earth is the same and equal tohe will experience the force of gravitywhereandis the mass of the Earth.
The difference between the two forces is small and reduces the weight of the man on the Equator slightly. If the man has a mass or 80 Kg, his weight at the pole will bebut his weight at the equator will be reduced by
]]>