rigid Bodies and Rotation

So far, you've probably only considered the effect a force has on linear motion. However, as I'm sure your intuition tells you, a force can also make an object rotate. For example, the flicking motion of the wrist when a frisbee is thrown, which gives the frisbee an angular acceleration. Or when a football is kicked, and it spins.

The Centre of Mass

The centre of mass of an object is the point on the object from which there is an equal amount of mass in every direction. When a force is applied directly to the centre of mass, only linear acceleration can occur. The centre of mass can be computed with an integral, or with a summation, but it's not necessary to know it for the IB. The centre of mass doesn't necessarily have to be on the object. For example, will a hollow ring, the centre of mass will always be in the centre if the ring

Rotational Equivalents of Linear Motion

The laws describing rotational motion are very similar to the laws describing linear motion. Most of the mathematical quantities used to describe linear motion have rotational equivalents.

Linear Equivalent	Rotational Equivalent	Description of the rotational equivalent
Displacement	Angular displacement	The change in angle. One revolution is 2pi radians or 360 degrees. Represented with the symbol theta, and measured in radians.
Velocity	Angular velocity	The derivative of angular displacement with respect to time. $ω = \frac{d θ}{d t}$ or when it's constant $ω = \frac{Δ θ}{Δ t}$ Measured in radians per second. Rads*s^-1
Acceleration	Angular acceleration	The second derivative of angular displacement with respect to time, and the first derivative of angular velocity with respect to time. Measured in radians per second squared. Rads*s^-2 ^{$α = \frac{d^{2} θ}{d t^{2}} o r α = \frac{d ω}{d t} . α = \frac{Δ ω}{Δ t} w h e n c o n s \tan t$}
Force	Torque	The "turning effect" of a force applied to an object. The larger the torque, the larger the change in angular acceleration. Torque is the cross product of the force vector and the displacement from the centre of mass. This is intuitive because if the force acts straight on the centre of mass, only linear acceleration will occur. $τ = F \times r o r τ = \|F\| \|r\| \sin θ$ . Theta is obviously the angle between the two vectors. There is also a rotational equivalent of Newton's second law (F=ma). The net torque can also be calculated in a similar way. It is the product of the momentum of inertia and the angular acceleration. $τ = I α$
Mass	Moment of Inertia	The resistance to angular acceleration due to torque. The larger the moment of inertia of an object, the less angular acceleration produced by a given torque. It is due to the distribution of mass, and it depends on the shape and density of an object. It is calculated as the sum of the product of the mass of every particle, and the distance from the particle to the axis of rotation. $I = Σ (m_{p} {r^{2}}_{p})$
Momentum	Angular momentum	The angular momentum of a system always remains constant, unless a net torque is acting on the system. Because it is a conserved quantity, it's incredibly useful. Angular momentum is calculated as the product of the momentum of inertia, and the angular velocity of rotation. $L = I ω$

Moment of Inertia and Conservation of Angular Momentum

The moment of Inertia of an object defined as the sum of this expression below for every single particle.

$m_{p} {r_{p}}^{2}$

m_prepresents the mass of the particle, and r_p represents the distance of the particle from the axis of rotation.

This expression tells us a lot about the moment of inertia of a rotating object.

Firstly, the axis you rotate it around matters. This is because the r_p² value will change for most particles when the axis of rotation is changed.

Secondly, if the mass is distributed further from the axis of rotation, the object becomes harder to rotate, as it has a larger moment of inertia. This is why divers bring their legs in when they do front flips.

Thirdly, if the mass is larger, the object becomes harder to rotate.

The fourth thing is when you want to find the moment of inertia of a complicated shape, you can break it down into simple shapes, and add their moments of inertia about the axis of rotation.

Angular Momentum

Editors

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