How can a torque on a mass push it forward? I was considering how a kayak travels through water, and the thought processes I came up with drove me to this question. Here's what went through my head (keep in mind, what went through my head about how a kayak works could be totally wrong, but nonetheless it's what I thought):


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*A person with  two oars in his/her kayak can be considered as one collective mass.

*Since this is a collective mass, if the person turned the boat with his left oar, there is a restoring force pushing the system in the exact opposite direction of the oar's force vector, which is towards the person, so that the water's restoring force vector is away the person.

*Since the system is a collective mass, this causes a torque on the mass, causing it to rotate in the positive direction. This makes sense, since the boat turns to the right if you drive the left oar. 


This made sense to me, since the farther you drive the oar from you, the easier you rotate (which should make sense since torque is directly proportional to the length of the radial vector).
However, this, in my view, shouldn't drive the boat forward, but merely cause it to rotate. Even when you use the left and right oar one after the other like one would do to go forwards, in my mind this would merely make it turn left, then right all in place, each time an oar is driven. 
In my view, only a force that is placed on the object's center of mass can cause it to move linearly instead of causing a torque and rotate it. However, this seems to not be true in reality. How should I expand my view of this, and my knowledge of torques, to explain how applying a torque can accelerate an object linearly (albeit not with as much force as it would if it were applied at the center of mass, I'd assume)?
Oh, and just as an aside, when I say "drive the oar" it's just a for-lack-of-a-better-term of using the oar to push the boat forward. I guess "row" is synonymous. 
 A: As I read this site to learn I don't have the ability to comment, but as a sea kayaker I wanted to clarify on the basic aspects of paddling.
The primary forces produced by a basic forward stroke is not an arc.  To obtain a reasonable amount of efficiency you put your blade in the water and pull the boat past the paddle in the most linear fashion possible.  While the force vector will be offset from the center-line and will induce yaw that is mostly balanced by by the following stroke.
It requires rotating your torso and leveraging both hands, pushing with the upper and pulling with the lower.
Your post most closely resembles what is called a sweep stroke which is use to turn in place.
The hull design of sea kayaks are typically quite similar to any displacement hull and will oppose athwartship and yaw forces, but as human's are very limited in power production most of the efficiency is ensuring that the strokes force is in the fore/aft axis as possible.
A: There are two possible combinations of driving the oar. Both of which provide forward motion (linear). Here's how:
Combination 1 (Both kayakers drive the oar at the same moment and to the same extent): No net torque. Only a forward push. Perfect linear motion (considering no whirls and eddies formed in water).
Combination 2 (Both kayakers drive the oar one after another, to the same or different extents, i.e. the most realistic way): The first drive/stroke has a forward force but at a distance from the center of mass, inducing a torque. So, the kayak goes forward and rotates at the same time. The trajectory is a cycloid. The next stroke (in the counter direction to the first stroke), causes a cycloidal trajectory that tries to correct the deviation caused by the first stroke. Thus, in average, the kayak maintains its forward (linear) trajectory only having such short deviations (that cancel out, in average).
P.S. : Your notion and thinking of how it works is correct, but it only goes half the way. You just needed to extend your imagination to consider the second stroke and the cycloidal (planar) motion. Think of two semicircles (one pointing up, and the other inverted) which touch each other at the circumference and have their diameters aligned along a straight line. Something like a sine wave starting from 0 rad to π rad (with no phase difference).
