A long string is wrapped around a 6.2cm diameter cylinder, initially at rest, that is free to rotate on an axle. The string is then pulled with a constant acceleration of 1.9 m/s^2 until 1.0 m of string has been unwound. If the string unwinds without slipping, what is the cylinder's angular speed, in rpm, at this time?
angular acceleration= 1.9/.062 rad/sec
wf=angacceleration*time
time can be found by 1m=1/2 1.9 t^2
To find the cylinder's angular speed in rpm, we need to first determine its linear speed, and then convert it to rotational speed.
To find the linear speed of the cylinder, we can use the formula:
v = at,
where:
v is the linear speed of the cylinder,
a is the acceleration of the string, and
t is the time taken to unwind the string.
First, let's find the time taken to unwind the string. We can use the equation of motion:
s = ut + 0.5at^2,
where:
s is the distance the string has been unwound (1.0 m),
u is the initial velocity of the string (which is 0 m/s since it starts from rest), and
a is the acceleration of the string (1.9 m/s^2).
Rearranging the equation, we get:
t^2 = (2s / a).
Substituting the values, we have:
t^2 = (2 * 1.0) / 1.9,
t^2 = 1.0526,
t ≈ √(1.0526),
t ≈ 1.026 s (rounded to three decimal places).
Now, let's find the linear speed of the cylinder using the formula mentioned earlier:
v = at,
v = 1.9 * 1.026,
v ≈ 1.9484 m/s (rounded to four decimal places).
Next, to convert the linear speed to rotational speed, we need to use the relationship between linear speed and angular speed. Since the cylinder has a diameter of 6.2 cm, its radius is half of that, or 3.1 cm (0.031 m).
The formula to relate linear speed (v) and angular speed (ω) is:
v = ωr,
where:
v is the linear speed of the cylinder,
ω is the angular speed of the cylinder, and
r is the radius of the cylinder.
Substituting the values, we have:
1.9484 = ω * 0.031,
ω = 1.9484 / 0.031,
ω ≈ 62.84 rad/s (rounded to two decimal places).
To convert the angular speed to rpm (revolutions per minute), we use the conversion factor:
1 revolution = 2π radians.
So, the angular speed in rpm is:
angular speed (rpm) = (ω / 2π) * 60,
angular speed (rpm) = (62.84 / 2π) * 60,
angular speed (rpm) ≈ 600 (rounded to the nearest whole number).
Therefore, the cylinder's angular speed, when 1.0 m of string has been unwound, is approximately 600 rpm.