How Does Torque Affect Gear Movement on a Rack?

[risecolt]

In this example I've designed a mechanism in which a gear is exposed to a linear and planar motion on a rack, on the top plane in the x and z coordinates. One degree of freedom has been eliminated, disabling it from moving in the y coordinates. Consider that the gears are masless.
This is a harmonic motion in an ideal system where friction is neglected. The gear rotates as the motor pin spins, but the planes of the motor remains constantly parallel to the flat plane of the rack. I need to calculate what the required torque is in order for the motor to conduct this motion.

If I use the equation for angular momentum, I could calculate the torque which is the change of angular momentum; M (torque) = w * I / s = (angular velocity * moment of inertia) / seconds. I wanted to accelerate this mass from an initial velocity of 0 RPM up to 60 RPM in one second. The motor has a mass of 1 kg and the gear has a radius of 25 mm.

Using the equation from the link below I have calculated the following:
(2pi/rad = 60RPM = angular velocity)

I = m/2 * (r^2 + (2l^2) = m/2 * (r^2 + 2*r^2) = m/2 * (3r^2) = 3/2 * m * r^2
M = 2pi * I / s = 0,00589 kgm^2/s^2 = 0,00589 Nm.

http://www.caddisegni.com/upload/calcoli/1-2.jpg

http://cognitivenetwork.yolasite.com/resources/Gear%20and%20Rack.png

Is this correct? Do you have a better solution?

Questions regarding:
http://www.caddisegni.com/upload/calcoli/1-2.jpg

Does this seem correct to you? If this equation can not be used for the example above, can it be used for the example in the link below?

http://cognitivenetwork.yolasite.com/resources/Gear.png

 

[LightHero]

No, this equation cannot be used for the second example. The equation from the link above is for calculating the torque needed to rotate a mass by a certain angular velocity in a certain amount of time. In the second example, you do not need to calculate the torque as the gear is already rotating and the motion is linear. Instead, you would need to calculate the force required to move the gear along the linear path, using a different set of equations.