Uniform circular motion

In uniform circular motion, the direction of the acceleration (the centripetal acceleration) is toward the center of the circule and its magnitude is a_c = v² / r, where v is velocity and r is radius of the circle.

In uniform circular motion: magnitude of velocity v = 2Πr / T, where r is radius and T is period (in seconds). Period T = 1/f, where f is frequency.

a_c = 4Π²r / T², where r is radius and T is period.

In unitform circular motion, the centripetal force is directed toward the center and maintains the circular motion; the so-called "centrifugal force" is actually not a force but the effect of the object's inertia to resist acceleration.

In uniform circular motion, the centripetal force does not do any work, since it only changes the direction of the velocity.

Center of Mass

Center o fmass is the point in a system about which all the mass of an object is concentrated. It is the dot that represents the object of interest in a free-body diagram.

For a homogeneous object where the density of mass is uniform, the center of mass is the object's geometric center.

For an object that is not homogeneous in mass, you can consider it as consisting of a collection of particles, then use the following method to caclulate its center of mass:

Step 1 Arbitrarily assign an origin point (0, 0, 0), and draw the x-axis, y-axis, and z-axis. (If all the particles are aligned in the same line, then no need for y-axis and z-axis; if all particles are aligned in the same plane, then no need for z-axis)

Step 2 Calculate the (x, y, z) coordinates for each particle

Step 3 Calculate the coordinates for the center of mass:

x_cm = (m₁x₁ + m₂x₂ + m₃x₃ + ... + m_nx_n)/(m₁ +m₂ +m₃ + ... + m_n)

y_cm = (m₁y₁ + m₂y₂ + m₃y₃ + ... + m_ny_n)/(m₁ +m₂ +m₃ + ... + m_n)

z_cm = (m₁z₁ + m₂z₂ + m₃z₃ + ... + m_nz_n)/(m₁ +m₂ +m₃ + ... + m_n)

Torque and moment arm distance

Torque is a measure of how much a force acting on an object causes that object to rotate. The object rotates about an axis, which we will call the pivot point or fulcrum, and will label 'O'. We will call the force 'F'. The distance from the pivot point to the point where the force acts is called the moment arm, and is denoted by 'r'. Note that this distance, 'r', is also a vector, and points from the axis of rotation to the point where the force acts. 

Torque is defined as

 = r x F = r F sin(θ).

In other words, torque is the cross product between the moment arm distance vector (the distance from the pivot point to the point where force is applied) and the force vector, 'θ' being the angle between r and F. 

Rotational Equilibrium

There may be more than one force acting on an object, and each of these forces may act on different point on the object. Then, each force will cause a torque. The net torque is the sum of the individual torques.

Rotational Equilibrium is analogous to translational equilibrium, where the sum of the forces are equal to zero. In rotational equilibrium, the sum of the torques is equal to zero. In other words, there is no net torque on the object.

Note that the SI units of torque is a Newton-metre, which is also a way of expressing a Joule (the unit for energy). However, torque is not energy. So, to avoid confusion, we will use the units N.m, and not J. The distinction arises because energy is a scalar quanitity, whereas torque is a vector. 

Moment of Inertia

In translational motion, linear inertia is the mass of the object.

In rotational motion, rotational inertia or moment of inertia is the mass times the square of perpendicular distance to the rotation axis, I = mr². 

If an object consists of multiple mass particles, the moment of inertia I =  ∑_im_ir_i² for all the point masses that make up the object.

Let’s take a simple example of two masses at the end of a massless (negligibly small mass) rod and calculate the moment of inertia about two different axes. In this case, the summation over the masses is simple because the two masses at the end of the barbell can be approximated as point masses, and the sum therefore has only two terms.

In the case with the axis in the center of the barbell, each of the two masses m is a distance R away from the axis, giving a moment of inertia of

I₁=mR^₂+mR^₂=2mR^₂

In the case with the axis at the end of the barbell—passing through one of the masses—the moment of inertia is

I₂=m(0)²+m(2R)²=4mR²

From this result, we can conclude that it is twice as hard to rotate the barbell about the end than about its center.

Radian vs degrees

2∏ radian = 360^ο degrees

Angular frequency & angular speed vs ordinary frequency & ordinary speed

Angular frequency = angular speed. Both refer to the magnitude of angular velocity. Both refer to the angular displacement per unit time. Their unit is radian / second.

Ordinary frequency f is the number of occurrences of a repeating event per unit of time. It is the reciprocal of period (unit is Hz: 1 Hz = 1 s ^-1).

Ordinary speed in uniform circular motion is the magnitude of the tangent velocity. It is the displacement along the circular path per unit time. Its unit is meter / second.

Angular frequency ω = 2πf, where f is ordinary frequency.

Angular speed ω = v/r, where v is tangent speed, and r is radius of the circle.

Comparison of Linear and Angular / Rotational Motion

You do not have to remember the angular/rotational equations. You can remember the linear equations and follow the following 3 rules to convert them to the corresponding angular equations:

• Convert displacement x (in meters) to angle (in radians)
• Convert mass inertia (mass, in kg) to rotational inertia (in kg*meter²)
• Convert force (in newton) to torque (in newton meter)

Angular momentum equations

Equation 1: L = Iω, where L is angular momentum, I is rotational inertia, and ω is angular velocity

Equation 2: L = rmv, where r is distance from the pivot point to the point where the force acts, m is mass, and v is linear velocity

The second equation can be proved as follows:

F = Δp/Δt = Δ(mv) / Δt

r*F = Δ(rmv) / Δt

Δ(rmv) = r*F*Δt

Notice r*F is torque, therefore r*F*Δt is angular momentum.