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Chapter no 4:
Center of mass
This toy uses the principles of center of mass to keep balance on a finger
In physics, the center of mass of a distribution of mass in space is the unique point where
the weighted relative position of the distributed mass sums to zero, or the point where if a force is
applied it moves in the direction of the force without rotating. The distribution of mass is balanced
around the center of mass and the average of the weighted position coordinates of the distributed
mass defines its coordinates. Calculations in mechanics are often simplified when formulated with
respect to the center of mass. It is a hypothetical point where entire mass of an object may be
assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle
equivalent of a given object for application of Newton's laws of motion.
In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body
has uniform density, it will be located at the centroid. The center of mass may be located outside the
physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In
the case of a distribution of separate bodies, such as the planets of the Solar System, the center of
mass may not correspond to the position of any individual member of the system.
The center of mass is a useful reference point for calculations in mechanics that involve masses
distributed in space, such as the linear and angular momentum of planetary bodies and rigid body
dynamics. In orbital mechanics, the equations of motion of planets are formulated as point
masses located at the centers of mass. The center of mass frame is an inertial frame in which the
center of mass of a system is at rest with respect to the origin of the coordinate system.
Definition
The center of mass is the unique point at the center of a distribution of mass in space that has the
property that the weighted position vectors relative to this point sum to zero. In analogy to statistics,
the center of mass is the mean location of a distribution of mass in space.
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Locating the center of mass
Plumb line method
The experimental determination of the center of mass of a body uses gravity forces on the body and
relies on the fact that in the parallel gravity field near the surface of the earth the center of mass is
the same as the center of gravity.
The center of mass of a body with an axis of symmetry and constant density must lie on this axis.
Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis
of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant
density is at the center of the sphere. In general, for any symmetry of a body, its center of mass will
be a fixed point of that symmetry.
In two dimensions
An experimental method for locating the center of mass is to suspend the object from two locations
and to drop plumb lines from the suspension points. The intersection of the two lines is the center of
mass.
The shape of an object might already be mathematically determined, but it may be too complex to
use a known formula. In this case, one can subdivide the complex shape into simpler, more
elementary shapes, whose centers of mass are easy to find. If the total mass and center of mass
can be determined for each area, then the center of mass of the whole is the weighted average of
the centers. This method can even work for objects with holes, which can be accounted for as
negative masses.
A direct development of the planimeter known as an integraph, or integerometer, can be used to
establish the position of the centroid or center of mass of an irregular two-dimensional shape. This
method can be applied to a shape with an irregular, smooth or complex boundary where other
methods are too difficult. It was regularly used by ship builders to compare with the
required displacement and center of buoyancy of a ship, and ensure it would not capsize.
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Applications
Engineering designs
Automotive applications
Engineers try to design a sports car so that its center of mass is lowered to make the
car handle better, that is maintaining traction while executing relatively sharp turns.
The characteristic low profile of the U. S. military Humvee was designed in part to allow it tilt farther
than taller vehicles, without a rollover, because its low center of mass would stay over the space
bounded the four wheels even at angles far from the horizontal.
Aeronautics
The center of mass is an important point on an aircraft, which significantly affects the stability of the
aircraft. To ensure the aircraft is stable enough to be safe to fly, the center of mass must fall within
specified limits. If the center of mass is ahead of the forward limit, the aircraft will be less
maneuverable, possibly to the point of being unable to rotate for takeoff or flare for landing.[18] If the
center of mass is behind the aft limit, the aircraft will be more maneuverable, but also less stable,
and possibly so unstable that it is impossible to fly. The moment arm of the elevator will also be
reduced, which makes it more difficult to recover from a stalled condition.[19]
For helicopters in hover, the center of mass is always directly below the rotorhead. In forward flight,
the center of mass will move forward to balance the negative pitch torque produced by
applying cyclic control to propel the helicopter forward; consequently a cruising helicopter flies
"nose-down" in level flight.
Astronomy
Two bodies orbiting their barycenter (red cross)
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The center of mass plays an important role in astronomy and astrophysics, where it is commonly
referred to as the barycenter. The barycenter is the point between two objects where they balance
each other; it is the center of mass where two or more celestial bodies orbit each other. When
a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting around a point that
lies away from the center of the primary (larger) body.For example, the Moon does not orbit the
exact center of the Earth, but a point on a line between the center of the Earth and the Moon,
approximately 1,710 km (1,062 miles) below the surface of the Earth, where their respective masses
balance. This is the point about which the Earth and Moon orbit as they travel around the Sun. If the
masses are more similar, e.g., Pluto and Charon, the barycenter will fall outside both bodies.
Body motion
Estimated center of mass/gravity of a high jumper doing a Fosbury Flop. Note that it is below the bar in this
position.
When high jumpers perform a "Fosbury Flop", they bend their respective bodies in such a way that
they clear the bar while their respective centers of mass do not necessarily do so.[22] Because it is the
height of the center of gravity (rather than of the highest part of the body) that constrains the
minimum energy investment for "clearing" the bar, "snaking over" the bar can reduce the energy
expended in propelling the body upward.
In kinesiology and biomechanics, the center of mass is an important parameter that assists people in
understanding their human locomotion. Typically, a human’s center of mass is detected with one of
two methods: The reaction board method is a static analysis that involves the person lying down on
that instrument, and use of their static equilibrium equation to find their center of mass; the
segmentation method relies on a mathematical solution based on the physical principle that
the summation of the torques of individual body sections, relative to a specified axis, must equal the
torque of the whole system that constitutes the body, measured relative to the same axis.
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Hydraulic system:
A hydraulic drive system is a quasi-hydrostatic drive or transmission system that uses
pressurized hydraulic fluid to power hydraulic machinery. The term hydrostatic refers to the transfer
of energy from pressure differences, not from the kinetic energy of the flow.
A hydraulic drive system consists of three parts: The generator (e.g. a hydraulic pump), driven by
an electric motor or a combustion engine or a windmill; valves, filters, piping etc. (to guide and
control the system); and the actuator (e.g. a hydraulic motor or hydraulic cylinder) to drive the
machinery.
Principle
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Principle of hydraulic drive system
Pascal's law is the basis of hydraulic drive systems. As the pressure in the system is the same, the
force that the fluid gives to the surroundings is therefore equal to pressure × area. In such a way, a
small piston feels a small force and a large piston feels a large force.
The same principle applies for a hydraulic pump with a small swept volume that asks for a
small torque, combined with a hydraulic motor with a large swept volume that gives a large torque. In
such a way a transmission with a certain ratio can be built.
Classification
Hydraulic drives are traditionally divided into three classes. These are:
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Industrial hydraulics.
Mobile hydraulics
Aircraft hydraulics
The classification is basically due to the fact that components are classified in these categories,
although some overlap exists between industrial and mobile hydraulics, aircraft hydraulics
components are highly specialized due to extreme requirements on weight and certification