I was so confused about this topic and watched this video the morning of my test and took notes.I ended up getting a 17.5/26 but then she curved the test since most people misread the word problems so 17.5/22 good enough for me 😬
Many people wonder why radians do not appear when we have radians*meters. Here is an attempt at an explanation: Let s denote the length of an arc of a circle whose radius measures r. If the arc subtends an angle measuring β = n°, we can pose a rule of three: 360° _______ 2 • 𝜋 • r n° _______ s Then s = (n° / 360°) • 2 • 𝜋 • r If β = 180° (which means that n = 180, the number of degrees), then s = (180° / 360°) • 2 • 𝜋 • r The units "degrees" cancel out and the result is s = (1 / 2) • 2 • 𝜋 • r s = 𝜋 • r that is, half of the circumference 2 • 𝜋 • r. If the arc subtends an angle measuring β = θ rad, we can pose a rule of three: 2 • 𝜋 rad _______ 2 • 𝜋 • r θ rad _______ s Then s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r If β = 𝜋 rad (which means that θ = 𝜋, the number of radians), then s = (𝜋 rad / 2 • 𝜋 rad) • 2 • 𝜋 • r The units "radians" cancel out and the result is s = (1 / 2) • 2 • 𝜋 • r s = 𝜋 • r that is, half of the circumference 2 • 𝜋 • r. If we take the formula with the angles measured in radians, we can simplify s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r s = θ • r where θ denotes the "number of radians" (it does not have the unit "rad"). θ = β / (1 rad) and θ is a dimensionless variable [rad/rad = 1]. However, many consider θ to denote the measure of the angle and for the example believe that θ = 𝜋 rad and radians*meter results in meters rad • m = m since, according to them, the radian is a dimensionless unit. This solves the problem of units for them and, as it has served them for a long time, they see no need to change it. But the truth is that the solution is simpler, what they have to take into account is the meaning of the variables that appear in the formulas, i.e. θ is just the number of radians without the unit rad. Mathematics and Physics textbooks state that s = θ • r and then θ = s / r It seems that this formula led to the error of believing that 1 rad = 1 m/m = 1 and that the radian is a dimensionless derived unit as it appears in the International System of Units (SI), when in reality θ = 1 m/m = 1 and knowing θ = 1, the angle measures β = 1 rad. In the formula s = θ • r the variable θ is a dimensionless variable, it is a number without units, it is the number of radians. When confusing what θ represents in the formula, some mistakes are made in Physics in the units of certain quantities, such as angular speed. My guess is that actually the angular speed ω is not measured in rad/s but in (rad/rad)/s = 1/s = s^(-1).
Where I get confused are the units. I'll be given 2 very basic units such as degrees per min and the radius, and be expected to find revolutions per hour. That makes figuring out whether I have angular speed or linear speed to be a pain.
You can treat units just like variables. Degrees per min might look like a fractions with degrees on top and minutes in the bottom. You can then convert these fractions by multiplying by 1 (unit fractions). Here is an example (1 degree) / (20 minutes) *. (60 minutes) / (1 hour) The “minutes” will cancel and then you can just work with the numbers giving us a result of (3 degrees) / (1 hour)
Can you please make a tutorial vedio about how to solve a Relationships between the linear and angular measures of a central angle in a unit center with word problems and solution please 🥺🥺🥺🙏🙏
