We can measure angles in degrees or radians. When studying angular motion, we will tend to use radians. If you are looking at a portion of a circle, the angle in radians can be calculated by dividing the arc length by the radius of the circle. The arc length for a complete circle is just the circumference, which can be calculated as 2πr (π represents the greek letter pi, which numerically is approximately 3.1416). A complete circle also represents an angle of 360°. From this we can get the conversion factor that 2π radians (or 6.28 radians) is the same as 360°.
a.) What is a 82.9° angle in terms of radians? (Normally you would enter units of rad, but in this case do not enter units.)
b.) If a wheel turns around 6.00 complete rotations, what would the angular displacement of that wheel be in radians? (Enter units of rad)
a. 6.28/360o = x/82.9o, x = 82.9 * 6.28/360 = 1.45 Radians.
b. 6.28rad/rev * 6revs =
a.) A 82.9° angle in terms of radians can be calculated as follows:
82.9° * (2π radians / 360°) ≈ 1.449 radians
b.) If a wheel turns around 6.00 complete rotations, the angular displacement of that wheel can be calculated as follows:
6.00 rotations * 2π radians/rotation ≈ 37.699 radians
a.) To convert degrees to radians, we can use the conversion factor that 2π radians is the same as 360°. Therefore, we can set up the following proportion:
2π radians = 360°
x radians = 82.9°
We can cross multiply and solve for x:
x = (82.9° * 2π radians) / 360°
x ≈ 82.9° * (2π/360°)
x ≈ 1.4485 radians
So, a 82.9° angle is approximately 1.4485 radians.
b.) If a wheel turns around 6.00 complete rotations, we need to calculate the angular displacement in radians. One complete rotation is equal to 2π radians, so:
Angular displacement in radians = (6.00 rotations) * (2π radians/rotation)
Angular displacement in radians = 12π radians
Therefore, the angular displacement of the wheel would be 12π radians.
a) To convert degrees to radians, we can use the conversion factor that 2π radians is equal to 360°.
So, to find the angle in radians, we divide the given angle in degrees by 360° and multiply it by 2π.
For a 82.9° angle, the calculation would be:
angle in radians = (82.9° / 360°) * 2π
Plugging in the values and simplifying:
angle in radians = (82.9 / 360) * 6.28
angle in radians ≈ 1.525
Therefore, the 82.9° angle is approximately 1.525 radians.
b) To find the angular displacement in radians, we need to know the number of complete rotations made by the wheel.
Given that the wheel turns around 6.00 complete rotations, we know that one complete rotation is equal to 2π radians.
So, to find the angular displacement in radians, we can multiply the number of rotations by 2π.
angular displacement in radians = number of rotations * 2π
For 6.00 complete rotations:
angular displacement in radians = 6.00 * 2π
angular displacement in radians = 12π
Therefore, the angular displacement of the wheel is 12π radians.