if force acts on a wheel away from the centre does it have both translatory and rotatory motion?( the wheel is not fixed.) And as per my knowledge,if force acts away from the line of axis it produces same acceleration as it produces when it is acting through the centre,so if in the 1st case if it produces rotation (due to torque) from where does the extra energy is getting to it??
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If the magnitudes of the forces are the same in the two cases, the linear acceleration would not be the same. The work done by the force would exactly equal to the sum of the linear energy and the rotational energy. Therefore its linear acceleration cannot be as much if it rotates as well.
If you prefer a quantitative result, you can decompose the force in two orthogonal components, one of which passes the center of mass. The component passing the center of mass would result in linear acceleration, while the other one would result in angular acceleration.
WunderNatur
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Assume that the force $F$ acts at the centre of mass of the wheel and this moves the centre of mass of the wheel a distance $d$.
The work done by the force is $Fd$ and this is the gain in translational kinetic energy of the wheel.
Now suppose that the force $F$ acts at some point on the wheel which is not its centre of mass and again the centre of mass moves a distance $d$.
$Fd$ will again be the gain in translational kinetic energy of the wheel.
However now as well as the centre of mass undergoing a translational motion the wheel acquires a rotational motion about its centre of mass.
This means that the point of application of the force must move a distance greater than $d$ so the work done by the force is now greater than when the force was applied at the centre of mass of the wheel.
That extra work done results in a gain in the rotational kinetic energy of the wheel.
Farcher
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