My whirlpool dryer recently stopped working where if u press the start button it tries to start up but does not. If u hold the button will try to start but will try for 10 seconds and stop. I have replaced the start button but it did not help. I was able to get running after spinning the drum but up am unable to get started again. Please let me know if you know what my problem may be.

Here are your parts
Parts for Whirlpool LER3634EW1 Dryer - AppliancePartsPros.com

See the attachment for the wiring diagram.

Try turning the drum in the correct direction.
It should turn fairly easily.
You may want to try this on a couple other units to get a feel for it.
This checks if the drum has too much drag on it so it cannot start.

If it is OK then odds are that the motor needs replacing.

Do you still get a hum?
Also is the hum like the motor trying to start but for some reason cannot.

Or is it that the motor is running but the drum not turning? In this case the belt is probably broken.

If it is now silent it could be that the thermal fuse on the blower assembly has blown. This kills power to the motor. Unplug the unit and check it. It should be 0 ohms.

The reason it tried to run and then shut off is that the motor has an internal thermal protect.
So if it overheats the protect opens killing power to the motor. Could be that this protect is not closing now. Unpluge the unit and check the motor windings with a meter. If either one is infinite ohms then the motor is toast.

The motor has two centrifugal switches on it.
They close when the motor gets close to operating speed.
One connects in the heater. This ensures that the heater does not come on till there is air flow.
The other has two functions. It provides a parallel path around the start switch. This lets you release the start switch and the unit keeps running.
It also switches out the motor's start winding which if left on would overheat the motor.
There may be a problem with these switches.
Niote that they are not available as a separate part.