A 79 diameter wheel accelerates uniformly about its center from 110 to 360 in 4.5 .
1. Determine its angular acceleration.
2.Determine the radial component of the linear acceleration of a point on the edge of the wheel 1.3 after it has started accelerating.
3.Determine the tangential component of the linear acceleration of a point on the edge of the wheel 1.3 after it has started accelerating.
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To solve these problems, we need to use the equations of rotational motion. There are a few key equations we can use:
1. Angular acceleration (α) = (change in angular velocity (ω)) / (change in time (t))
2. Radial component of linear acceleration (ar) = (angular acceleration (α)) * (radius (r))
3. Tangential component of linear acceleration (at) = (change in tangential velocity (v)) / (change in time (t))
Let's solve each problem step by step:
1. To determine the angular acceleration, we need to use the formula:
Angular acceleration (α) = (change in angular velocity (ω)) / (change in time (t))
Given:
Initial angular velocity (ω1) = 110 degrees
Final angular velocity (ω2) = 360 degrees
Time (t) = 4.5 seconds
First, let's convert the angular velocities from degrees to radians by multiplying by π/180:
ω1 = 110 * π/180 radians
ω2 = 360 * π/180 radians
Now let's calculate the angular acceleration:
α = (ω2 - ω1) / t
= (360 * π/180 - 110 * π/180) / 4.5
= (250 * π/180) / 4.5
≈ 4.36 radians/second^2
Therefore, the angular acceleration is approximately 4.36 radians/second^2.
2. To determine the radial component of the linear acceleration of a point on the edge of the wheel 1.3 seconds after it has started accelerating, we need to use the formula:
Radial component of linear acceleration (ar) = (angular acceleration (α)) * (radius (r))
The diameter of the wheel is given as 79 inches, so the radius (r) would be half of that, which is 79/2 inches.
Given:
Angular acceleration (α) = 4.36 radians/second^2
Radius (r) = 79/2 inches
Time (t) = 1.3 seconds
Let's calculate the radial component of linear acceleration:
ar = α * r
= 4.36 * (79/2)
≈ 171.57 inches/second^2
Therefore, the radial component of the linear acceleration of a point on the edge of the wheel 1.3 seconds after it has started accelerating is approximately 171.57 inches/second^2.
3. To determine the tangential component of the linear acceleration of a point on the edge of the wheel 1.3 seconds after it has started accelerating, we need to use the formula:
Tangential component of linear acceleration (at) = (change in tangential velocity (v)) / (change in time (t))
The tangential velocity can be calculated using the formula:
Tangential velocity (v) = (angular velocity (ω)) * (radius (r))
Given:
Angular velocity (ω) = ω1 + α * t (using the initial angular velocity and angular acceleration from the first problem)
Time (t) = 1.3 seconds
Radius (r) = 79/2 inches
First, let's calculate the angular velocity:
ω = ω1 + α * t
= 110 * π/180 + (4.36) * (1.3)
= (110 * π/180) + (4.36 * 1.3)
≈ 110 * π/180 + 5.668
Now let's calculate the tangential velocity:
v = ω * r
= (110 * π/180 + 5.668) * (79/2)
≈ (110 * π/180 + 5.668) * 79/2
Finally, we can calculate the tangential component of linear acceleration:
at = (v - ω * r) / t
Now you can substitute the values of ω, r, and t into the equation to find the tangential component of linear acceleration.