You can make mobiles out of most anything, but the easiest materials to work with are paper, wood and foam. When constructing a mobile, pay attention to balancing points, particularly if you are using odd numbers of pieces or bars for them to hang from. The easiest method is to make one top piece and hang two or four objects evenly balanced from it, then add one or two things onto each of them, as far as you can go based on size. Also try linear or round mobiles for more looks depending on your preference.
Paper
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When using paper or cardboard for a mobile, you have more room for error because of the light weight of the materials. You can draw and cut out your own shapes, or find plenty of shapes online. Color and cut out the objects you want to include in your mobile. You can leave them flat or make two sides for each object and stuff them with paper or tissue before stapling or gluing the two sides together for a more three-dimensional mobile. Punch holes in the top of the objects and attach string or yarn to them. Make the base for your mobile by unbending wire coat hangers or using stiff wire. Cut the wire into the lengths you want the bars of your mobile to be, and loop the ends using needle-nose pliers, to make a place to hang the objects from. You can make an X of the two bars to start out with four objects hanging, one from each corner, or make a free-form mobile with several varying bars and objects. Tie on the objects where needed.
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Wood
You will want a sturdier base for a wood mobile, such as a wooden cross piece or a metal bar. Use picture hanging wire, fishing line or strong blended-material string to hang your pieces once they are cut. You can use paper templates as above, or use woodcutting templates. A popular material light enough for easy assembly yet durable and strong is to use thin boards, like a 1/8-inch thickness. Any wood will do, even plywood, but a hardwood will be easier to varnish or paint. If you use a rough wood, sand it down first with a hand sanding block. For any wood, sand down the edges of the shapes so there is less chance of getting scratches or splinters. Drill small holes where you want the objects to hang in the cross piece or bar, and knot the wire, line or string through the holes. Make a hole or pair of holes for the string to hang the mobile from, in the center of the base. Once your pieces are ready, sanded and painted or varnished, drill small holes in their tops and knot the other ends of the hanging line or wire through these. Note, when using wire, it may be easier to wrap than knot it.
Foam
For small children or infants' rooms, a hanging mobile is a great decoration that will interest a young child. Foam is the perfect medium for a kid's room or nursery because of its softness. You can use an unbent wire coat hanger or lengths of wire for the base, bending over the ends into loops using needle-nose pliers. Use precut foam shapes or cut your own from craft foam. Thread a heavy needle with fishing line and pierce several of the foam shapes onto one line for a dangling mobile, or hang them each separately if desired, poking a hole with a straight pin or needle to thread the line through. You can also use other lightweight objects as the base, such as twigs or chopsticks, or even colorful plastic straws.