An object initially rotating at an angular speed of 1.6 rad/sec turns through 30 revolutions during the time it experienced an angular acceleration of 0.32 rad/s^2.
a) For how much time did the acceleration last?
b) What was the final angular speed?
Given:
Initial angular speed (ω_initial) = 1.6 rad/sec
Number of revolutions (n) = 30
Angular acceleration (α) = 0.32 rad/s^2
a) To find the time the acceleration lasted (t), we can use the equation:
n = ω_initial * t + 0.5 * α * t^2
Since a revolution is equal to 2π radians, the initial angular speed can be converted to radians per second.
ω_initial = 1.6 rad/sec * 2π = 10.08π rad/sec
Now, substitute the values into the equation:
30 revolutions = 10.08π * t + 0.5 * 0.32 * t^2
Simplifying the equation:
15π = 5.04π * t + 0.16 * t^2
0.16t^2 + 5.04πt - 15π = 0
Unfortunately, there is no simple algebraic solution to this quadratic equation. It can be solved numerically using a calculator or software. The positive value of 't' obtained from the solution will be the time the acceleration lasted.
b) To find the final angular speed (ω_final), we can use the equation:
ω_final = ω_initial + α * t
Substitute the known values:
ω_final = 1.6 + 0.32 * t
Using the calculated value of 't' from part a), you can find the final angular speed.
To solve these questions, we can use the following formulas:
1) Angular displacement (θ) = (initial angular speed * time) + (0.5 * angular acceleration * time^2)
2) Final angular speed (ωf) = initial angular speed + (angular acceleration * time)
Let's use these formulas to find the answers:
a) For how much time did the acceleration last?
We are given the initial angular speed (ωi) = 1.6 rad/sec, the angular acceleration (α) = 0.32 rad/s^2, and the angular displacement (θ) = 30 revolutions.
First, we need to convert the angular displacement from revolutions to radians. Since there are 2π radians in one revolution, the angular displacement in radians is:
θ = 30 revolutions * 2π radians/revolution = 60π radians
Now, let's substitute these values into the equation:
θ = (ωi * t) + (0.5 * α * t^2)
60π = (1.6 * t) + (0.5 * 0.32 * t^2)
Rearranging the equation to solve for time (t), we get a quadratic equation:
0.16t^2 + 1.6t - 60π = 0
Using the quadratic formula, we can solve for time (t):
t = (-1.6 ± √(1.6^2 - 4 * 0.16 * (-60π))) / (2 * 0.16)
Calculating this expression, we get two values for time: t ≈ 16.832 seconds or t ≈ -5.332 seconds. Since time cannot be negative, we discard the negative solution.
Therefore, the acceleration lasted for approximately 16.832 seconds.
b) What was the final angular speed?
We can use the second formula:
ωf = ωi + (α * t)
Substituting the given values, we get:
ωf = 1.6 rad/sec + (0.32 rad/s^2 * 16.832 seconds)
Calculating this expression, we find that the final angular speed (ωf) is approximately 6.404 rad/sec.