ssm A string trimmer is a tool for cutting grass and weeds; it utilizes a length of nylon "string" that rotates about an axis perpendicular to one end of the string. The string rotates at an angular speed of 56 rev/s, and its tip has a tangential speed of 60 m/s. What is the length of the rotating string?
in m
Vt=rw
r=Vt/w
so:
r = 60m/s÷[56rev/s(2πrad/1rev)
r=0.17 m
To find the length of the rotating string, we need to use the formula for the circumference of a circle. In this case, the rotating string forms a circle with the radius equal to the length of the string.
The formula for the circumference of a circle is:
C = 2πr
Here, we are given the angular speed of the string, which is given in revolutions per second (rev/s). To find the circumference, we need to convert the angular speed from revolutions to radians.
Since 1 revolution is equal to 2π radians, we can multiply the angular speed by 2π to convert it to radians per second:
Angular speed in radians per second = 56 rev/s * 2π radians/revolution = 112π radians/s
We also know that the tangential speed of the string tip is 60 m/s. The tangential speed is the linear speed at the outer edge of a rotating object and is equal to the angular speed multiplied by the radius:
Tangential speed = Angular speed * Radius
In this case, the tangential speed is given as 60 m/s and the angular speed is 112π radians/s. Solving for the radius:
Radius = Tangential speed / Angular speed
= 60 m/s / 112π radians/s
≈ 0.536 m
The radius of the rotating string is approximately 0.536 m.
Finally, to find the length of the rotating string, we can use the circumference formula:
C = 2πr
= 2π * 0.536 m
≈ 3.37 m
Therefore, the length of the rotating string is approximately 3.37 meters.
To find the length of the rotating string, we can use the formula:
Length of string = Tangential speed × Time for one revolution
We are given the angular speed, which is 56 rev/s. Since one revolution is equivalent to 2π radians, the angular speed can be converted to radians per second as:
Angular speed = 56 rev/s × 2π rad/rev = 112π rad/s
Next, we need to find the time for one revolution. The time for one revolution can be calculated as the reciprocal of the angular speed:
Time for one revolution = 1 / angular speed = 1 / (112π) s
Finally, we can substitute the given tangential speed, which is 60 m/s, and the time for one revolution into the formula:
Length of string = Tangential speed × Time for one revolution
= 60 m/s × (1 / (112π) s)
= 60 / (112π) m
≈ 0.1689 m (rounded to four decimal places)
Therefore, the length of the rotating string is approximately 0.1689 meters.