A wheel turns with an angular velocity of 8 rad/s.
a. Express the angular velocity to the nearest tenth of a revolution per minute.
b. If the radius of a wheel is 20 cm, how far will it roll in 10 s, to the nearest centimeter?
a. 8 rad/s * (1 rev / 2 π rad) * 60 s/min
b. 8 rad/s * 20 cm/rad * 10 s
a. Wow, we're going from rad/s to revolutions per minute? That's like measuring speed in snail races instead of mph! Anyway, let's convert. 1 revolution = 2π radians. So, the angular velocity of 8 rad/s is approximately 8 / (2π) revolutions per second. To find revolutions per minute, we multiply by 60 since there are 60 seconds in a minute. Let me do the math... *buzzing noises* The angular velocity is approximately 76.4 revolutions per minute. Ain't that a spinning sensation?
b. Now we're rolling! So, the distance rolled by the wheel in 10 seconds would be equal to the circumference of the wheel multiplied by the number of revolutions it made. The formula is: distance = 2πr * number of revolutions. With a radius of 20 cm, the formula becomes: distance = 2π * 20 cm * number of revolutions. Just a moment while I crunch the numbers... *calculating noises* And voila! The wheel will roll approximately 400π cm in 10 seconds, which is approximately 1256 cm when rounded to the nearest centimeter. Wheely impressive, isn't it?
a. To express the angular velocity in revolutions per minute, we need to convert it from rad/s to rev/min.
Given:
Angular velocity = 8 rad/s
To convert from rad/s to rev/min, we can use the conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds
First, we need to find the number of revolutions in 1 second:
8 rad/s * (1 revolution / (2π radians)) = 8 / (2π) revolutions/second
Now, let's find the number of revolutions in 1 minute:
8 / (2π) * 60 seconds = (8 * 60) / (2π) revolutions/minute ≈ 76.4 revolutions/minute
Therefore, the angular velocity to the nearest tenth of a revolution per minute is approximately 76.4 revolutions per minute.
b. To find the distance the wheel will roll in 10 seconds, we need to calculate the circumference of the wheel and multiply it by the number of revolutions.
Given:
Radius of the wheel = 20 cm
Time = 10 s
The circumference of the wheel is given by the formula: Circumference = 2πr
Circumference = 2π * 20 cm ≈ 125.7 cm
To find the distance rolled in 10 seconds, we need to multiply the angular velocity in revolutions per second by the time:
Distance rolled = Angular velocity * Circumference * Time
= 8 rad/s * (1 revolution / (2π radians)) * 125.7 cm * 10 s
= 8 * 125.7 * 10 / (2π) cm
≈ 2513.33 cm ≈ 2513 cm (rounded to the nearest centimeter)
Therefore, the wheel will roll approximately 2513 centimeters in 10 seconds.
a. To express the angular velocity in revolutions per minute, we need to convert the angular velocity from rad/s to revolutions per minute.
To do this, we need to know the conversion factor between radians and revolutions. One revolution is equal to 2π radians.
Therefore, to convert from rad/s to revolutions per minute, we use the following conversion factors:
1 revolution = 2π radians
1 minute = 60 seconds
First, let's convert the angular velocity from rad/s to revolutions per second:
Angular velocity in revolutions per second = (angular velocity in rad/s) / (2π rad/revolution)
Angular velocity in revolutions per second = 8 rad/s / (2π rad/revolution) ≈ 1.2732 revolutions per second
Next, we can convert the angular velocity from revolutions per second to revolutions per minute:
Angular velocity in revolutions per minute = (angular velocity in revolutions per second) * (60 seconds/1 minute)
Angular velocity in revolutions per minute ≈ 1.2732 revolutions per second * 60 seconds/1 minute ≈ 76.392 revolutions per minute
Therefore, the angular velocity to the nearest tenth of a revolution per minute is approximately 76.4 revolutions per minute.
b. To find how far the wheel will roll in 10 seconds, we need to calculate the distance traveled by the circumference of the wheel.
The formula to calculate the distance traveled by the circumference of a wheel is:
Distance traveled = (angular velocity) * (radius of the wheel)
Distance traveled = (8 rad/s) * (20 cm)
Distance traveled = 160 cm
Therefore, the wheel will roll approximately 160 cm in 10 seconds, to the nearest centimeter.