1. A flywheel is rotating at 7 rad/sec. It has a 15-cm diameter. What is the speed of a point on its rim, in cm/min?
2. A wheel is rotating at 3 radians/sec. The wheel has a 30 cm radius. What is the speed of a point on its rim, in m/min?
3. What is the angular speed in radians/second associated with a rotating an angle 3pi/4 radians in 2 seconds?
speed is distance/time
the circumference is 2π times the radius, so that's how far a point on the wheel travels in one revolution. So, for problem
#1
7 rad/s * 1rev/(2π rad) * (π*15 cm/rev) = 7*π*15/(2π) = 105/2 cm/s
#2 is just the same
#3 (3π/4 rad)/(2s) = 3π/8 rad/s
1. First, we need to find the circumference of the flywheel's rim.
- The diameter of the flywheel is given as 15 cm, which means the radius is half of that, or 7.5 cm.
- The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
- Plugging in the values, we have C = 2π(7.5) = 15π cm.
2. Now, we can find the speed of a point on the rim in cm/min.
- The speed of a point on the rim can be calculated using the formula S = rω, where S is the speed, r is the radius, and ω is the angular speed.
- Given that the angular speed is 7 rad/sec and the radius is 7.5 cm, we have S = (7.5)(7) = 52.5 cm/sec.
- To convert cm/sec to cm/min, we can multiply by 60: S = 52.5 × 60 = 3150 cm/min.
3. To find the angular speed associated with a rotation of an angle, we use the formula ω = θ / t, where ω is the angular speed, θ is the angle in radians, and t is the time taken.
- Given that the angle is 3pi/4 radians and the time taken is 2 seconds, we have ω = (3pi/4) / 2 = (3/4)(pi/2) = 3pi/8 radians/second.
1. To find the speed of a point on the rim of a flywheel, we need to convert the angular speed from radians per second to linear speed in centimeters per minute.
First, let's find the circumference of the flywheel's rim. The diameter is given as 15 cm, so the radius would be half of that, which is 15/2 = 7.5 cm. The circumference is calculated using the formula C = 2πr, where r is the radius. Therefore, the circumference is 2π(7.5) = 15π cm.
To convert the angular speed from radians per second to linear speed in centimeters per minute, we multiply the angular speed (in this case, 7 rad/sec) by the circumference of the flywheel's rim.
Speed (in cm/min) = Angular speed (in rad/sec) * Circumference (in cm)
= 7 * 15π
= 105π cm/min
So, the speed of a point on the rim of the flywheel is 105π cm/min.
2. To find the speed of a point on the rim of a wheel in meters per minute, we need to follow a similar process as in the previous question.
First, let's find the circumference of the wheel's rim. The radius is given as 30 cm, so the circumference is calculated as C = 2πr, where r is the radius. Therefore, the circumference is 2π(30) = 60π cm.
To convert the angular speed from radians per second to linear speed in meters per minute, we multiply the angular speed (in this case, 3 rad/sec) by the circumference of the wheel's rim.
Speed (in m/min) = Angular speed (in rad/sec) * Circumference (in m)
= 3 * (60π/100) [Converting cm to m]
= 180π/100
= 1.8π m/min
So, the speed of a point on the rim of the wheel is 1.8π m/min.
3. The angular speed in radians/second associated with a rotating angle of 3π/4 radians in 2 seconds can be calculated using the formula:
Angular speed (in radians/second) = Angle (in radians) / Time (in seconds)
In this case, the angle is given as 3π/4 radians and the time is given as 2 seconds. Plugging these values into the formula:
Angular speed (in radians/second) = (3π/4) / 2
= 3π/8 radians/second
So, the angular speed associated with rotating an angle of 3π/4 radians in 2 seconds is 3π/8 radians/second.