The pendulum is powered by a weight suspended by a chain hanging from a sprocket wheel inside the clock. The chain is placed over the teeth on the wheel. As the wheel turns the chain is lowered tick by tick to its full length in the manner of a pulley system. Start the pendulum swinging with a gentle push by your finger. According to Wayne Berry, a certified clockmaker (Cuckoo Clock Repair Troubleshooting Guide), if the chain is properly engaged with the teeth on the sprocket wheel the pendulum will continue to swing. If not, the chain will slide right on through to its full length. The chain of your clock may have slipped off the wheel, stopping the pendulum.
There are other causes for a pendulum's movement to cease. John Barnette, a certified clocksmith in Tennessee, says the clock must be correctly placed on the wall in a vertical position. It is a good idea to use a carpenter's level to check the vertical alignment of the clock. If the clock is tilted either from front to back, or side to side, the chain may hang up as it moves through the hole at the bottom of the clock. If the hole is too small, or if it is off center, the chain is prevented from passing through. Widening the hole a little may solve the problem. If anything is preventing the sprocket wheel from turning, the pendulum won't move.
Sometimes the problem may be the case of the clock. Many cuckoo clocks in use today are very old. Over time a case could become warped, hindering the vertical movement of the chain as it drops. The World of Cuckoo Clocks reminds us that the pendulum may stop simply because the weight is obstructed by a chair or other piece of furniture placed too close to the wall beneath the clock. If the weight comes to rest on anything, the movement of the pendulum will stop. If the clock is hung too low on the wall, the weight will reach the floor too soon and stop the pendulum.
Your pendulum may have stopped simply because the clock is dirty. Clocks must be oiled now and then with clarified oil or clock lubricant. According to Bill's Clockworks, as time passes, the lubricant picks up dust and forms a gooey paste that is abrasive to the clock's working parts. When the clock is dirty the works of the clock may stick and prevent the pendulum from moving. Instead of risking damage to a fine precision instrument by doing any repairs or the cleaning job yourself, a better choice might be to take your clock to a professional clockmaker.