Torque

We know that force is fundamentally defined as the interaction an object experiences that causes its motion to change linearly. How does force translate to a rotational frame of reference? The answer is not nearly as different as you may think. Torque, also known as moment of force in some areas, is the rotational analog of force and when you exert a torque on an object, its rotational motion will change.

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Also, keep in mind that since the units for the moment of inertia are kg meters squared, the units for torque are N m(Newton-meters).

The equation for the torque an object creates is given by:

which is the cross product between the force and the radius at which the force acts(also known as the distance vector). If you haven't fully learnt what cross products are, the cross product is the vector that is perpendicular to both the force and the radius. This usually can be configured through the sine of the angle between the force and the radius but since there are rare cases where you may have to use cosine, this general rule of thumb is better to know.

Newton's First Law states that an object at rest or a constant linear velocity's motion won't change unless acted on by a net external force. This is exactly the same for an object in rotation, like a bike wheel. Newton's First Law in terms of rotating objects states that an object at rest or at a constant angular velocity's motion won't change unless acted on by a net torque. Let's take a bike wheel for example. If you stop pedaling on your bike, the wheel will stop rotating because of forces like aerodynamic friction applying torque on the wheel. However, if you pedal at a steady pace such that the torque you create by pedaling equals the torque acting against the wheel, the wheel will rotate undisturbed at a constant rate, or angular velocity.

Newton's Second Law(Linear) states that the net force on an object is the derivative of its linear momentum with respect to time. This can be expressed mathematically as:

The rotational analog states that the net torque on an object is the derivative of angular momentum with respect to time. This can be expressed mathematically as:

However, if we assume the mass to be constant for the linear case(a very fair assumption to make), the net force on an object can be defined as the product of an object's mass and its acceleration. This is the exact same for net torque on an object, which can be defined as the product of an object's moment of inertia and angular acceleration, written as:

Remember that for a rotating system, you can't define directions as simplistically as you do with linear systems. Instead of breaking positive and negative directions into right and left, respectively, for both x and y, you split up directions for vectors in rotation by counter-clockwise(positive) and clockwise(negative). Any torque that causes motion counterclockwise is considered positive while any torque that causes clockwise motion is considered negative. This convention is arbitrary but is the standard for most applications of rotational mechanics.

Newton's Third Law in linear motion simply states that if an object A exerts a force on another object B, B will exert a force of equal magnitude and opposite direction on A, also known as "every reaction has an equal and opposite reaction". For rotational motion, this is quite similar, as expected. This basically states that for every torque exerted on an object, the object experiences a torque in the opposite rotational direction of the same magnitude.