Energy and Momentum Methods for Rigid Bodies in Plane Motion

Newtons second law in determining kinetics relationships is not always
the most ef?cient, although it always works. As for particles, energy
and momentum methods are often useful to analyze
rigid bodies in plane motion.

Work of a Force on a Rigid Body

The work of a force acting on a rigid body moving from position 1 to 2 is

Work of a Moment

The work of a moment has a similar form, for angular positions ?,

In the common case where the moment vector M is perpendicular
to the plane of motion,
M ? d? = M d?.
It is important to note those forces that do no work:
1. Forces that act at ?xed points on the body do not do work.
For example, the reaction at a ?xed, frictionless pin does no work
on the body that rotates about that pin.
2. A force which is always perpendicular to the direction of the
motion does no work.
3. The weight of a body does no work when the bodys center
of gravity moves in a horizontal plane.
4. The friction force ^ at a point of contact on a body that rolls
without slipping does no work. This is because the point of contact
is the instantaneous center of zero velocity.

Kinetic Energy of a Rigid Body

The kinetic energy of a particle only consists of the energy associated
with its translational motion. The kinetic energy of a rigid body also
includes a term for the rotational energy of the body,

where C is the center of mass of the rigid body.
The kinetic energy of a rigid body rotating about an arbitrary
axis at point O is

Principle of Work and Energy

The principle of work and energy for a rigid body is the same
as used for particles with the addition of the rotational energy terms.

where  T₁  = initial kinetic energy of the body
T₂  = ?nal kinetic energy of the body
U₁₂  = work of all external forces and moments acting on the body
moving from position 1 to 2 This method is advantageous when
displacements and velocities are the desired quantities.
Conservation of Energy The conservation of energy in a conservative
rigid body system is

where  T  = kinetic energy
V  = total potential energy (gravitational and elastic)

Power

The net power supplied to or required of the system is

This can be calculated by taking time derivatives of the kinetic and
potential energy terms. Each term is considered positive when it
represents the power supplied to the system and negative when
power is taken from the system.

Impulse and Momentum of a Rigid Body

Impulse and momentum methods are particularly useful when time
and velocities are of interest. Figure 1.3.20 shows how rigid bodies
are to be considered for this kind of analysis. Notice that rotational
motion of the rigid body must be included in the modeling.

FIGURE 1.3.20 Impulse and momentum for rigid bodies.

The impulse of the external forces in the given interval is

where t is time, C is the center of mass, and ?F includes all
external forces.
The impulse of the external moments in the given interval is

For plane motion, if ?M is parallel to ?, the scalar expressions are

Impulse and Momentum of a System of Rigid Bodies

A system of rigid bodies can be analyzed using one of the two following
procedures, illustrated in Figure 1.3.21.
1. Apply the principle of impulse and momentum to each rigid
member separately. The mutual forces acting between members must be
included in the formulation of the solution.
2. Apply the principle of impulse and momentum to the entire system
of bodies, ignoring the mutual forces between members.

FIGURE 1.3.21 System of rigid bodies.

Conservation of Momentum

The principle of conservation of linear and angular momentum of
particles can be  extended to rigid bodies that have no external forces
or moments acting on them. The conservation of linear momentum
means that the center of mass C moves at a constant speed in
a constant direction,

Likewise, for conservation of angular momentum of rigid bodies,

For a system of rigid bodies, use the same ?xed reference point O for
all parts of the system. Thus, for plane motion,

There are two important points to remember when using these equations.
First, ?HC = 0 does not imply that ?HO = 0, or vice versa. Second,
conservation of momentum does not require the simultaneous
conservation of both angular and linear momenta (for example, there
may be an angular impulse while linear momentum is conserved).

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By : E-book Mechanical_Engineering_Handbook

Prefabricated ceiling tiles are attached either to an existing ceiling or to furring strips. Staples or nails, either with or without adhesive, or adhesive alone secures the tiles.

Often, ceiling tiles show the effect of water damage. To conceal stains or streaks in tiles, apply a primer or clear sealer. When the tiles are dry, you can paint them with latex paint.

Another problem you may en-counter is dents or chips in tiles. Illus-trated at right are the steps in removing and replacing a dented tile with tongue-and-groove edges. You cut through all four joints and pry the tile off its backing; pry out the cut-off tongues from the grooves in the adjacent tiles. Use pliers to remove remaining staples or nails; scrape off adhesive.

After cutting the tongue off one side of the replacement tile, apply ad-hesive to the back of the tile or to the
ceiling (follow the tile manufacturer’s directions). Position the tile over the opening, slip the remaining tongue into the groove of an adjacent tile, and press in place until the adhesive holds; or use a floor-to-ceiling brace to hold the tile until the adhesive is dry.

Replacing A Damaged Ceiling Tile

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Painting Over Wall Repairs

A fresh coat of paint is a fast and effective way to conceal wall repairs. Depending on the size of the repair and the availability of matching paint, you can paint just the repaired area, the whole wall, or the entire room. Here are some guidelines to help you get ready for the job.

Selecting paint and tools. The type of surface you’re covering determines the kind of paint required. If you have paint left over from the original job, you’re in luck. If not, refer to the chart at right for the appropriate paint.

Choosing the correct brush is im-portant, too. The type of bristle should suit the paint or stain you’re using; the brush size must fit the job. Select a natural-bristle brush to apply oil-base paint, Polyurethane, varnish, or shellac. Use brushes with synthetic bristles to apply water-base (latex) paint.

For wood stains, use either type of brush. Choose a 1-inch brush for hard-to-reach areas, a 2 to 3-inch brush for medium-size surfaces, and a 31/2 to 4-inch brush or 9-inch roller for large areas, such as entire walls. Use a thick-napped roller for textured walls.

Preparing the surface. Before you can apply the paint, you may need to sand and wash the surface. In most cases, you will at least have to apply a primer to ensure that your repair will not show. For more information, see the Sunset book Wall Coverings.

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Repairing Wood Floors
Replacing Wood Flooring
Silencing Squeaky Floors

A good wood floor will last the life of your home and actually improve with age. But even the best wood floors are subject to damage or other problems.

The problem may be in the surface flooring or related to defects in the supporting structure. To successfully repair the floor, it’s important to know what type of flooring you have and how it’s attached to the subfloor, as well as the cause of the damage or problem.

Flooring structure, types of wood flooring, and common problems are described below. On the following
pages you’ll find directions for repairing surface damage, fixing individual boards, replacing damaged sections of flooring, and silencing squeaky floors both from the surface and from underneath.

The subfloor may be constructed from 1 by 4, 1 by 6, or 2 by 6 lumber, or plywood panels. In a lumber subfloor, boards are laid diagonally across joists. A plywood subfloor has panels laid in a staggered fashion with the ends of the panels butted together over and nailed to the joists.

If your house is built on a concrete slab, the floor may be laid over wood 2 by 4s (called sleepers) and nailed, or laid on a base of plywood and fastened with nails or adhesive.

Types of wood floors. Two common types of wood flooring are strip and plank. Boards for strip flooring come in random lengths and uniform widths.(Common widths are I1/2, 2, 21/4, and 31/4 inches.) Plank flooring, a holdover from colonial days, was originally made from maple planks more than a foot wide. Today, it differs little from strip flooring, except that it comes in random widths (usually 3, 5, and 7 inches), as well as random lengths. In both types, board length usually ranges from 2 to 8 feet; most boards are 3/4 inch thick.

Both strip and plank boards may be milled with square or tongue-and-groove edges and ends, or with a combination of both. Depending on the milling, strip and plank floorings may be blind nailed, face nailed, or screwed to the subfloor (see drawing below).

Tongue-and-groove strip flooring is almost always blind nailed; square-edged strip flooring is usually face
nailed. Tongue-and-groove plank flooring may be blind nailed, screwed, or both; square-edged planks may be face nailed, screwed, or both. Plugs usually indicate that flooring is screwed to the subfloor, though plugs are some-times used just for decoration.

Common floor problems. Problems can range from minor surface damage to serious structural defects. Daily wear and tear on a wood floor causes surface scratches and gouges. The natural expansion and contraction of wood occurring with changes in temperature or humidity as well as alternate drying and wetting due to leaks, can cause nails to pull out of the wood, allowing flooring boards to separate or warp;
even the joists and the subfloor can separate. The natural settling of a house can also cause separations between the joists and the subfloor.

If your floor is sagging or uneven, there could be a serious structural problem; consult a professional.

Repairing the floor. You can repair most minor surface damage yourself (see facing page). If your floor has separated, split, loose, or warped boards, first determine the cause and correct it, if necessary; then make the repair (see facing page).

If the damage is too extensive for a simple repair, you may need to replace boards or sections of flooring as described. (Directions for replacing damaged sections depend on how the boards are milled and the way they’re secured to the subfloor, so be sure to make these determinations before going ahead.) To eliminate squeaks in floors, you can choose one of several methods, from lubricating the squeak with graphite to
installing wood shims.

Finally, if your floor’s overall ap-pearance and condition are suffering, you may want to make repairs and
replacements where necessary, then refinish the entire floor. For refinishing techniques, see the Sunset book Do-lt-Yourself Flooring.

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Repairing Wood Floors

Surface damage on wood floors, as well as separated, split, loose, or warped boards, can be successfully
repaired. When you’re refinishing the repaired area, match its color and pro-tective finish as closely as possible to the surrounding area to effectively hide the repair.

Repairing surface damage
To preserve the beauty of your wood floors, repair surface damage, such as water or burn marks, scratches, and gouges, immediately after it occurs. In most cases, you’ll first have to remove the protective wax or oil finish from the damaged area with a wax stripper.

Once you complete the repair, you must rewax or oil the area. When strip-ping and rewaxing, follow the manufacturer’s directions for the product you’re using.

Water marks. After removing the pro-tective finish, rub the marks with fine-grade steel wool and a little paste wax or a solvent-base liquid floor wax. If the marks don’t disappear, wipe the wax with a soft cloth and rub again with fine-grade steel wool and odorless mineral spirits. Wipe clean and finish. Burn marks. For burn marks that just darken the wood’s surface, lightly sand; wipe up sanding residue with a damp cloth. When dry finish as de-sired.

For deeper burns, carefully scrape out the burned wood with a sharp knife. Apply one or more coats of a commer-cial scratch hider, putty stick, or stick shellac; then finish.

Scratches and gouges. You can con ceal a shallow scratch with one or two. applications of a commercial
scratch hider or crayon. To repair deep scratches and gouges, remove wax or oil from the damaged area. Fill the scratch or gouge with matching wood putty putty stick, or stick shellac. Let dry; then sand smooth with fine-grade sandpaper and finish.

Fixing damaged floor boards
It’s best to repair separated, split, loose, or warped floor boards as soon as trouble appears.

For long or wide separations be-tween boards, fill the gaps with a wood strip (see below). For the best fit, angle the saw blade to cut a strip that tapers in slightly at the bottom.

Loose, split, or warped boards can be satisfactorily repaired, as shown below. Sand any putty when it’s dry
with fine-grade steel wool and finish the floor as desired.

Separated Boards

Split Boards

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Loose Boards

Warped Boards

Replacing Wood Flooring

If your damaged flooring doesn’t respond to the remedies, you may need to replace some boards. The job doesn’t require exceptional skill, but it does take patience and finesse.

Ideally replacing boards will be one step in an overall floor refinishing project. If not, you’ll have to take special care to match and finish the re-placement boards. To ensure a good match, take the old boards when you shop for replacements. With plank flooring, make a drawing of the dam-aged area showing the dimensions of the planks you’re removing.

Before replacing flooring, look for evidence of the nailing method. In blind nailing, nails are driven through each board’s tongue, and they don’t show from the finished surface. Dots of wood putty indicate face nailing. Plugs on plank flooring often cover screws; if the plugs are just decorative, the flooring is blind nailed (some flooring secured with screws may be blind nailed as well). For any of these, you can use one
of two approaches.

One approach, shown below, is to remove the damaged boards in a stag-gered pattern. This produces a less no-ticeable repair and is best for an open floor area. The other approach (see facing page), good for areas that will be covered by a rug or furniture, is to cut out a rectangle, remove the damaged boards, and replace them.

Illustrated here are directions for replacing blind-nailed tongue-and-groove boards. You’ll follow the same
general sequence to replace other types of flooring, but because they’re attached differently to the subfloor, note special instructions that follow. Face-nailed square-edged boards. Use a nailset to drive the nails through the boards and into the subfloor. If you’re using the rectangular pattern, cut only the ends of the boards. Starting at one end, use a prybar to remove them. With the staggered pattern, follow Steps 1-3 below.

Lay new boards (cut to size) in place; use a hammer and 2 by 4 wood block to tap the last one into place.
Face nail boards to the subfloor, matching the nail pattern in the original flooring. Countersink the nail heads with a nailset and hammer; fill the holes with wood putty After the putty is dry sand it smooth and finish the area to match the existing flooring.

Plugged planks. After marking the area to be cut, use an electric drill to drill out the wood plugs down to the screw.

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Replacing Tongue-And-Groove Boards (Staggered)

heads; then remove the screws. If your planks are also face nailed (look for dots of wood putty), use a nailset and hammer to drive nails through the planks and into the subfloor.

To cut a rectangular pattern, set the blade of a power circular saw to the thickness of the planks and cut the four sides, using a wood guide strip; care-fully remove the planks with a prybar. To remove planks in a staggered pat-tern, follow Steps 1-3 (facing page).

Mark the replacement planks and cut them to length. Using the plugs in the existing planks as guides for spacing, mark the locations of screw holes on the ends of each replacement plank and on the ends of each existing plank adjacent to the opening. At each location, use an electric drill with combination bit to drill a hole for the plug and a pilot hole for the screw shank.

The diameter and depth of the holes should be the same as those in the existing flooring.

Lay the planks one at a time. Insert and tighten the screws. Daub glue in the plug holes and seat the plugs,
removing excess glue. When the glue is dry sand the plugs flush with the surface and finish the boards to match the existing flooring.

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Replacing Tongue-And-Groove Boards (Rectangular)

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Silencing Squeaky Floors

It’s often relatively simple to eliminate floor squeaks once you pinpoint the cause. Knowing why floors squeak and how to determine the cause of the squeaks you hear will help you choose the appropriate repair method illustrated below.

What causes squeaks. The squeaks you hear when you walk across a floor occur when pieces of wood rub to-gether. Squeaks in wood floors can originate in the finished floor, subfloor, joists, bridging, or other parts of the supporting structure. Common causes are separations between the joists and subfloor (due to drying, inadequate nailing, or settling), weak or loose bridging, and ill-fitting or warped floor boards.

Locating squeaks. In a house where joists are visible from the basement or crawl space, you’ll be able to pinpoint squeaks more easily than in homes where joists aren’t exposed or where it’s an upper floor that’s making the noise.

If joists are exposed, watch from below while another person walks across the floor above; you should be
able to detect the probable cause of the squeak. For example, you may spot slight movement between joists and the subfloor, or loose bridging between joists. If the joists are not exposed, you’ll have to confine your investigations to the finished flooring.

Correcting squeaks. Simple remedies include squirting powdered or liquid graphite between boards or dusting cracks between boards with talcum powder. You can also apply floor oil to the floor or a few drops of mineral oil between boards. (Use mineral oil sparingly; too much can stain the surface of the floor.)

If the squeaks are coming from ill-fitting or warped boards, you can often fix them from above. Work wood putty between boards or try one of the reme-dies below. Nail through the flooring from above only when you can’t work from below; the areas where you nail may be somewhat conspicuous.

In a home where joists are exposed, you can work on the floor from underneath. Toenail loose bridging; tighten other loose areas as shown below.

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Three Ways To Silence Squeaks From Above

Three Ways To Silence Squeaks From Below

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