BASIC Introduction: Torque and Rotation
Introduction
Rotational motion focuses on an object revolving on a fixed axis. This article will briefly cover up the key topics related to rotation. Before we begin with that, we will be focusing on the basics: angular velocity(ω) and rotational acceleration. Angular velocity is equal to the displacement of θ per unit time(Δθ/Δt) and is measured in the units radians per second. Rotation acceleration is equal to the rate of change of angular acceleration(α). In order to convert between a rotational form and a linear form, the rotational form must be multiplied by r.
Torque Formula: τ = rFsinθ
Torque is the force applied to an object changing its rotation. The formula for torque depends on three things: the force, the radius, and the axis. The force will clearly effect torque. Radius affect torque as well since an object will not rotate if force is applied on its point of rotation. Finally sinθ or the axis has an impact on torque as the angle of the force impacts its strength with a perpendicular force being the strongest. Imagine opening a door, the most optimal way to open a door is perpendicularly and the farthest from the hinge(the axis of rotation).
To better understand this concept we will be solving the 1.249 Irodov problem:
A uniform disc of radius R is spinned to an angular velocity ω and then placed on a horizontal surface. How long will the disk be rotating on the surface if the friction coefficient is k? The pressuere exerted by the disc on the surface can be assumed to be uniform.
We know that the reason the disc will stop is because of the torque. The torque though is not the same at every point in the circle. It increases with distance. This means we can select a small ring, find the torque and then integrate to find the total torque. Using the torque, you are able to find the angular acceleration, which allows you to figure out how long the disc will rotate. The frictional force on the ring will be k * dN.
dτ = k * dN * r
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dτ = k * dmg * r
dτ = k (2pi * r & dr)/ (pi * R²) * mg r
dτ = (k * 2 * r² * dr * mg)/R²
τ = (2 * k * mg)/ R² * integral from 0 to R of r² dr = 2/3 K mg R
Torque is equal to α * I, where I represents moment of inertia(an objects resistance to motion).
α = (4kg)/(3R) now using the rotational motion equations:
0 = (ω — αt) … we can solve for t.
We get t to be: (3ωR)/(4kg)
v = rω
Imagine a carousel; the entire carousel is spinning at the same angular velocity, as both the horses on the outside of the carousel and the inside of the carousel complete the full rotation in the same amount of time, but the linear velocities are different. This is because the horses on the outside cover a larger distance in the same amount of time. The scenario described helps to explain where the equation, from above, comes from. Linear velocity is directly proportional to radius and angular velocity, while radius and angular velocity are inversely proportional.
