A wheel's uniform angular acceleration is constant. Initially its angular velocity is zero. During the first 1.0 s time interval, it rotates through an angle of 86.0°.
a) Through what angle does it rotate during the next 1.0 s time interval?
(b) Through what angle during the third 1.0 s time interval?
To solve this problem, we can use the equations of rotational motion. The equation relating angular displacement, initial angular velocity, angular acceleration, and time is:
θ = ω_i * t + (1/2) * α * t^2
Where:
θ = angular displacement (in radians)
ω_i = initial angular velocity (in radians per second)
α = angular acceleration (in radians per second squared)
t = time interval (in seconds)
For the first time interval, given:
θ = 86° = 86 * (π/180) radians,
ω_i = 0 (since the initial angular velocity is zero),
α = constant.
(a) To find the angle during the next 1.0 s time interval, we need to calculate the final angular velocity at the end of the first interval. The final angular velocity can be found using:
ω_f = ω_i + α * t
Since ω_i = 0, we have:
ω_f = α * t
Using this final angular velocity, we can find the angular displacement during the next 1.0 s time interval using the equation mentioned above. Let's calculate it:
ω_f = α * t = α * (1.0 s)
θ_next = ω_f * t + (1/2) * α * t^2
θ_next = (α * (1.0 s)) * (1.0 s) + (1/2) * α * (1.0 s)^2
Simplifying:
θ_next = α * (1.0 s) + 0.5 * α * (1.0 s)^2
θ_next = α + 0.5 * α
Therefore, the angle rotated during the next 1.0 s time interval is:
θ_next = 1.5 * α
(b) Similarly, to find the angle during the third 1.0 s time interval, we can use the same equation:
θ_third = ω_i * t + (1/2) * α * t^2
Since ω_i = 0, the equation simplifies to:
θ_third = (1/2) * α * t^2
Substituting the given values:
θ_third = (1/2) * α * (1.0 s)^2
θ_third = 0.5 * α
Therefore, the angle rotated during the third 1.0 s time interval is:
θ_third = 0.5 * α
To determine the angle of rotation during the next time intervals, we need to use the equations of motion for rotational motion. The key equation for this scenario is the kinematic equation for angular displacement, which relates the initial angular velocity, the angular acceleration, and the time:
θ = ω_initial * t + (1/2) * α * t^2
where:
- θ is the angle of rotation
- ω_initial is the initial angular velocity
- α is the angular acceleration
- t is the time interval
Now let's find the answers to the questions:
(a) Through what angle does it rotate during the next 1.0 s time interval?
Given:
- Initial angular velocity (ω_initial) = 0 rad/s
- Angular acceleration (α) is constant
Using the equation θ = ω_initial * t + (1/2) * α * t^2, we can substitute the given values:
θ = 0 * 1 + (1/2) * α * 1^2
θ = (1/2) * α
Therefore, during the second 1.0 s time interval, the wheel rotates through an angle of (1/2) times the angular acceleration.
(b) Through what angle during the third 1.0 s time interval?
Since the angular acceleration is constant, the angle of rotation during each time interval will be the same. Therefore, during the third 1.0 s time interval, the wheel will also rotate through an angle of (1/2) times the angular acceleration.