A gear with a radius of 32 centimeters is turning at pi/9 radians/sec. What is the linear speed (velocity) at a point on the outer edge of the gear?
since the arc length is radius * angle,
arc speed (linear speed) is radius times angular speed:
32 * pi/9 cm/s
Well, isn't this a wheel-y interesting question! To find the linear speed at a point on the outer edge of the gear, we need a pinch of trigonometry and a dash of clown flair!
The circumference of a gear can be calculated using the formula C = 2πr, where r is the radius. In this case, the radius is 32 centimeters, so the circumference is approximately 2π(32) = 64π centimeters.
Since the gear rotates at pi/9 radians per second, we can use the formula linear speed = angular speed × radius to find the linear speed. Plugging in the values, we have linear speed = (pi/9) × 32 = 32π/9 centimeters per second.
So, at a point on the outer edge of the gear, the linear speed is approximately 32π/9 centimeters per second. That's pretty rad, isn't it?
To find the linear speed at a point on the outer edge of the gear, you will use the formula:
Linear speed = Angular speed * Radius
Given:
Radius (r) = 32 centimeters
Angular speed (ω) = π/9 radians/sec
Substituting the given values into the formula:
Linear speed = (π/9) * (32 cm)
To simplify the expression, you can approximate the value of π to 3.14:
Linear speed ≈ (3.14/9) * (32 cm)
Simplifying further:
Linear speed ≈ 3.54 cm/sec
Therefore, the linear speed at a point on the outer edge of the gear is approximately 3.54 cm/sec.
To find the linear speed (velocity) at a point on the outer edge of the gear, you need to know the formula connecting angular speed and linear speed.
The formula is:
Linear speed = Angular speed * Radius
Given that the gear has a radius of 32 centimeters and is turning at π/9 radians/sec, you can substitute these values into the formula to calculate the linear speed.
Linear speed = (π/9) * 32
To simplify this, you can multiply π/9 by 32 to get the value in centimeters per second.
Linear speed ≈ 3.56 centimeters/second
Therefore, the linear speed at a point on the outer edge of the gear is approximately 3.56 centimeters/second.