A gyroscope slows from an initial rate of 32.0 rad/s at a rate of 0.700 rad/seconds squared. How many revolutions does it make before stopping?
If only I could find the angular velocity first, in radians/second...
I'm assuming 0.700 rad/second squared is angular acceleration, and so far I have only been taught equations to find angular acceleration with time, and I don't know time.
How do I find angular velocity given this information? Once I get there, I can convert it to revolutions.
Huh? Initial angular velocity is given at 32 radians/sec. If you want it in rev/sec, divide by 2PI. But you dont have to do that.
finalw=0=32rad/sec -.7rad/sec^2*t
time=32/.7 seconds
displacement= avgw*time= 16rev/sec*32/.7 sec
revs= displacement in radians/2PI
To find the final angular velocity, you can use the equation:
Final Angular Velocity (ωf) = Initial Angular Velocity (ωi) + (Angular Acceleration (α) * Time (t))
However, since you don't have the time, you need to find another way to solve the problem. In this case, you can use the fact that the gyroscope stops when its final angular velocity becomes zero.
So, you can set the final angular velocity (ωf) to zero and solve for time (t):
0 = ωi + (α * t)
Rearranging the equation, you have:
t = - ωi / α
Now, you can substitute the given initial angular velocity and angular acceleration into the equation:
t = - 32.0 rad/s / (-0.700 rad/s^2)
Simplifying the expression, you get:
t = 45.71 s
Now that you have the time, you can find the angular displacement (θ) using the equation:
θ = ωi * t + (1/2) * α * t^2
Plugging in the values:
θ = (32.0 rad/s) * (45.71 s) + (1/2) * (-0.700 rad/s^2) * (45.71 s)^2
Simplifying the expression, you get:
θ ≈ 643.45 rad
Now, to find the number of revolutions, you need to convert this angular displacement to revolutions.
Since 2π radians is one revolution, you can use the conversion factor:
1 revolution = 2π radians
Therefore, the number of revolutions is:
Number of Revolutions = θ / (2π)
Plugging in the value of θ, you get:
Number of Revolutions = 643.45 rad / (2π) ≈ 102.34 revolutions
Therefore, the gyroscope makes approximately 102.34 revolutions before stopping.