a wheel starts from rest and rotates with a constant angular acceleration and reaches an angular speed of 12 rad/sec in 3 sec. find the angle through which it rotates in this time.
Averate speed is 6 rad/sec, over a time of 3 seconds
displacement= avgspeed*time
To find the angle through which the wheel rotates, we can use the formula:
θ = ωi * t + (1/2) * α * t^2
Where:
θ is the angle through which the wheel rotates (in radians),
ωi is the initial angular speed of the wheel (in radians per second),
t is the time taken for the wheel to reach the final angular speed (in seconds),
and α is the constant angular acceleration of the wheel (in radians per second squared).
In this case, the initial angular speed is 0 (because the wheel starts from rest), the final angular speed (ωf) is 12 radians per second, and the time taken (t) is 3 seconds.
The formula becomes:
θ = 0 * 3 + (1/2) * α * 3^2
θ = 0 + (1/2) * α * 9
θ = (1/2) * α * 9
Now, from the given information, we know that the wheel reaches an angular speed of 12 radians per second in 3 seconds. We can use this information to find the constant angular acceleration (α).
We can use the formula to find α:
ωf = ωi + α * t
Substituting the given values:
12 = 0 + α * 3
12 = 3α
Dividing both sides of the equation by 3:
4 = α
Now, we can substitute α = 4 into the formula for θ:
θ = (1/2) * α * 9
θ = (1/2) * 4 * 9
θ = 2 * 9
θ = 18 radians
Therefore, the wheel rotates through an angle of 18 radians in 3 seconds.