a wheel revolves
about a fixed horizontal axis through O.
During a certain interval of time, the
angular velocity (which initially is 20 rad
per second clockwise) changes uniformly
at such a rate that the angular
displacement is 60 rad counterclockwise,
and the total angle turned through is 100
rad. Point A is in the position shown 2.5
sec after the beginning of the time
interval.
Determine, for this instant, the linear
acceleration of point A.
To determine the linear acceleration of point A, we need to understand the relationship between angular velocity and linear acceleration in a rotating object.
The linear acceleration of a point on a rotating object can be calculated using the formula:
a = r * α
Where:
a is the linear acceleration,
r is the distance between the point and the axis of rotation (in this case, the distance of point A from the axis of rotation),
and α is the angular acceleration (the rate at which the angular velocity changes).
In this problem, we are given the initial angular velocity, the angular displacement, and the total angle turned through the interval of time. We need to find the angular acceleration, and then we can use the formula to calculate the linear acceleration.
Step 1: Find the angular acceleration:
The angular acceleration can be calculated using the formula:
α = Δω / Δt
Where:
α is the angular acceleration,
Δω is the change in angular velocity,
and Δt is the change in time.
In this problem, the initial angular velocity is given as 20 rad/s clockwise, and the angular displacement is 60 rad counterclockwise.
To convert the initial angular velocity to counterclockwise motion, we change the sign of the velocity:
Initial angular velocity = -20 rad/s (counterclockwise)
The total angle turned through is given as 100 rad.
Now we can calculate the change in angular velocity and change in time:
Δω = final angular velocity - initial angular velocity
Δω = 0 rad/s - (-20 rad/s) = 20 rad/s
Δt = angular displacement / Δω
Δt = 60 rad / 20 rad/s = 3 s
Therefore, the angular acceleration is:
α = Δω / Δt = 20 rad/s / 3 s = 6.67 rad/s² (counterclockwise)
Step 2: Calculate the linear acceleration of point A:
We need to know the distance between point A and the axis of rotation. Since it is not given in the problem, we cannot calculate the linear acceleration directly. Additional information is needed.
Please provide the distance from point A to the axis of rotation, and I will assist you further in calculating the linear acceleration.