Page 3 of 6

When we take the second derivative with respect to time, we get a function having units of velocity divided by time or acceleration. But before we look at the acceleration function for circular motion, let's get straight the concepts of centrifugal acceleration and centripetal acceleration. This will allow us to better understand what the acceleration function is describing.

When an object is traveling in a circle at a constant speed. A centripetal force is continuously acting on the object pulling it inward along the radius towards the center source, which counters the object's momentum to travel in a straight line. It's this tendency to travel straight that creates the effect of an outward push that the object experiences; but this effect is not the centrifugal force. The actual centrifugal force is equal and opposite the centripetal force, and exerts a pull on the center source, which comes from the object in motion trying to travel in a straight line.

So the centripetal force acts on the object in motion, pulling it inward and keeping it from traveling in a straight line; while the centrifugal force acts on the center source, pulling it outward as the object tries to travel straight.

Now in the case of a washer's spin cycle. The washer's drum exerts a push against the wet clothes, pushing them inward, and keeping them from traveling straight. While the clothes exert an equal and opposite push against the drum, as they try to travel in a straight line. So depending on the circular motion setup, this interaction between opposing forces can take the form of push rather than a pull, but the action-reaction relationship remains the same.

A couple final points to make. Centrifugal force and centripetal force are equal in magnitude and opposite in direction. But centrifugal acceleration and centripetal acceleration are not equal in magnitude. They are defined by:

Accel = Mutual Force ÷ mass

So it's incorrect to think that an object maintains its circular motion by balancing the centripetal acceleration with the centrifugal acceleration. Circular motion requires an inward pull or centripetal acceleration in order to counter its tendency to travel straight. Providing this exact acceleration is what maintains circular motion.

And finally, an acceleration acting on an object traveling perpendicular to the acceleration, will only effect the objects direction and not its speed. This is because the acceleration vector doesn't have a component that aligns with the object's direction of motion.


Comments

Post a Comment

Popular posts from this blog

Page 6 of 6

Below is a list of some measurable effects that emerge from circular motion, but in fact are not physically real: 1) Reactance in ac circuits from oscillating periodic motion. It impedes the flow of ac current, but is not a physical property of the circuit element producing it like resistance is. 2) Time dilation from the circular flow of time; by making the speed of light invariant, you are saying that the total flow of motion through spacetime is c, whether it's motion in the time dimension, the spatial dimensions, or both; this total flow of motion is the circular flow of time following spacetime curvature. 3) The effect in lump ac circuits that allows you to view the sinusoidal source as instantaneously traveling through the circuit—even though the wavelength is much longer than the closed loop circuit path it travels. 4) The outward push that an object in circular motion experiences which is mistakenly taken as the centrifugal force; this is an effect from the momentum of the ...
Read more

Page 4 of 6

Now let's look at the acceleration function for circular motion, which is the second derivative of the rotational function with respect to time: F(t)ᵣₒₜ = Re^(iwt) F'(t)ᵥₑₗ = iwRe^(iwt) F''(t)ₐ = -w²Re^(iwt) The units here are R/w² which is dist/sec² for acceleration. We notice that the function is being multiplied by the -1 operator. Which means the acceleration vector is opposite the original rotational vector on the real axis. When we take the tail end of the acceleration vector and attach it to the front tip of the rotational vector; we notice that the acceleration points inward towards center as the rotational vector traces a circle. This then is the centripetal acceleration which acts on an object in circular motion, keeping it from traveling in a straight line tangent to the circle. Because of Newton's third law of reactionary forces, the centripetal acceleration causes an equal and opposite acceleration called the centrifugal acceleration which is a real for...
Read more

Page 5 of 6

The imaginary unit i=√-1, is not a real number, it is a math construct that intertwines with real numbers in the complex number plane. This intertwining produces real measurable effects in the physical world without in fact being real. The first derivative of the exponential function in Euler's formula reveals that circular motion is encoded with the imaginary unit i, here it represents a direction perpendicular to the circle's radius at every point on the circle. So linear motion is actually the limiting case of circular motion. Where you get piecewise linearity for very small sections of the circle(sect→0). And linearity over larger sections as the radius increases(r→∞). This is why the kinetic energy formula for linear motion applies to circular motion as well: Krot = ½Iw² Where: w is angular velocity and I is moment of inertia  A point mass m at radius r has a moment of inertia(I): I = mr² and v = wr Krot = ½mr²w² = ½mv² By keeping the encoded imaginary unit i, we get: K...
Read more