50 hz motor at 60 hz

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well you can but syncronous motors use the mains frequency to control their speed. the motor will run fast.
There are a couple of things you could do, Build a 50hz power supply or alter the motor pully output size to match the new speed.

What is the use, record player motors often had alternative pulleys or idler wheels to cope with frequency differances in different parts of the world for example with out changing the motor.
 
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What is the application and is it speed sensitive?

Motors designed to run at 50Hz but operating at 60Hz will generally run cooler but about 20% fast which will matter if the motor is used in a speed sensitive application such as in a turntable or capstan motor in a tape deck. (Fans, supply and take up reel motors don't care, but need the right value of phase shift cap installed if present.)

Motors wound for 60Hz and running on 50Hz often have issues with overheating and need to be derated due to additional losses imposed due to insufficient winding inductance.

Motors using phase shift capacitors (larger synchronous motors) will need different value capacitors in order to maintain the proper phase shift for quiet operation and good torque output. 50Hz motors with proper value phase shift caps do not need to be derated when run on 60Hz providing that they are running within the allowable design voltage range.
 
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