Introduction
The primary purpose of activities such as swinging and rocking is to stimulate your baby's vestibular system, and thereby improve her sense of balance.
Activities good for stimulating the vestibular system include:
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Rocking your baby
When your baby is newborn, you can cradle her in your arms and rock her. You can also hold her against your chest while you sit on the edge of a seat and rock your body back and forth.
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Swinging your baby
When your baby can hold her head steady, you can start swinging her left and right, forward and backwards. Initially, you can swing her by holding her by your hands under her arms close to the armpits (she is facing away from you). Some baby jumpers (those that hang your baby by an elastic band to the top of the door frame) can also be modified and used for these purposes.
With a partner, you could also lay your baby on a blanket, and have you and your partner hold each end of the blanket (each hand holding each blanket edge) and swing your baby in the blanket.
When your baby is physically much stronger (usually closer to her first birthday), you can also swing her by letting her sit with her upper thighs on your hands (your hands clasping her thighs tight) and her back leaning against your forearms. This is like she were sitting in a playground swing.
At that stage, you can also look for playground swings with baby seats instead of just a flat plank.
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Spinning your baby
The easiest way to spin your baby is to hold her against your chest or in a cradle position and spin yourself around. Make sure you keep your balance and don't get yourself too dizzy!
When your child is older, you can also spin her while holding her upper arms. However, do make sure you have enough room and her legs do not hit anything!
Acceleration
This is what many fathers like doing - "throwing" the baby - except you don't literally have to throw your baby to get the required effect. Hold your baby (facing you) under the armpits, start with her stationary on the ground, and then lift her up quickly to as high a position as you can without letting her go.
Upside-down
This is an excellent exercise for stimulating your baby, which unfortunately is not commonly done. If you are just starting out, you may wish to have your partner assist you by making sure your baby's head and neck are not placed in an awkward position at any time.
The simplest way is to sit on a chair and let your baby lie on her back on your lap, with her head close to your knees and her legs closer to you. Next, lift up your baby's legs and hold on to her ankles (or hold further up just under the knees if you feel you need more control). Then slowly stand up as you hold on to your baby's legs. You need not even do anything with your hands except hold on to your baby, and when you have fully stood up, you will find yourself now holding your baby upside-down. To lie her back down in a horizontal position, simply reverse the process - hold her close to your thighs, and slowly sit back down as you gently ease her into a horizontal position onto your lap.
In the beginning, keep these upside-down periods short (5 seconds at a time) until your baby becomes more used to it. As you and your baby get more comfortable, you can extend the period to a minute, and even gently swing her.
Obviously, do not do this immediately after your baby has had her meal.