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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6$\begingroup$ Possible duplicate of Force applied off center on an object $\endgroup$– Hritik NarayanCommented Jul 5, 2017 at 15:29
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2 Answers
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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.
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$\begingroup$ @akhil If the force acts through the axis, one of the components in the decomposition above is zero, so there's only translation. If the force acts away from the axis, neither components would be zero. Therefore there are both translation and rotation, but the translational acceleration would be smaller than the 'through the axis' case. $\endgroup$ Commented Jul 5, 2017 at 15:46
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$\begingroup$ For example the wheel is at rest in space and force acts on it away from axis will it have both translatory and rotatory motions? And if force acts on a wheel parallely how can u reslove them explaning roatory motion? $\endgroup$– akhilCommented Jul 5, 2017 at 15:47
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$\begingroup$ @akhil yes for the first question. I don't understand the second one. $\endgroup$ Commented Jul 5, 2017 at 15:50
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$\begingroup$ How can the wheel be rotated if only one force acts on it? the spinning action of the wheel takes place only when two opposite forces acting on top and bottom of wheel right???( like applying forces on tap) $\endgroup$– akhilCommented Jul 5, 2017 at 16:05
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$\begingroup$ @akhil This is not true. One force is enough to generate rotation, as long as there's torque (that is, the force doesn't pass through the center of mass). Imagine if the force is always perpendicular to the radius passing through the point where the force acts on, there can be only rotation but no translation! $\endgroup$ Commented Jul 5, 2017 at 16:15
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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.
