A Ferris wheel with radius 8.8 m rotates at a constant rate, completing one revolution every 34.6 s. Suppose the Ferris wheel begins to decelerate at the rate of 0.212 rad/s2 when a passenger is at the top of the wheel. Find the magnitude and direction of the passenger's acceleration at that time.
There will be backward tangential acceleration at a rate 0.212 rad/s^2*R, and downward acceleration of V^2/R. Gravity is not accelerating him unless he jumps off.
Add the two accelerations vectorially. They are perpendicular.
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The fastest measured pitched baseball left the pitcher's hand at a speed of 50.0 . If the pitcher was in contact with the ball over a distance of 1.50 and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it?
To find the magnitude and direction of the passenger's acceleration, we need to calculate the centripetal acceleration and the tangential acceleration separately and then combine them.
Centripetal acceleration is given by the formula:
ac = (v^2)/r
where v is the velocity and r is the radius of the Ferris wheel.
First, we need to find the angular velocity (ω), which is the rate of change of angle with respect to time.
ω = (2π)/T
where T is the time taken for one complete revolution.
Substituting the given values:
ω = (2π)/(34.6 s) = 0.182 rad/s
Next, we find the velocity (v) using the formula:
v = ωr
Substituting the given values:
v = (0.182 rad/s)(8.8 m) = 1.6016 m/s
Now, we can calculate the centripetal acceleration (ac):
ac = (v^2)/r
ac = (1.6016 m/s)^2 / 8.8 m
ac ≈ 0.2916 m/s^2
Next, we find the tangential acceleration (at):
at = α r
where α is the deceleration rate and r is the radius of the Ferris wheel.
Substituting the given values:
at = (-0.212 rad/s^2) (8.8 m)
at ≈ -1.8656 m/s^2
Finally, we can combine the centripetal acceleration (ac) and the tangential acceleration (at) vectorially using the Pythagorean theorem:
a = √(ac^2 + at^2)
a = √(0.2916 m/s^2)^2 + (-1.8656 m/s^2)^2
a ≈ 1.882 m/s^2
The magnitude of the passenger's acceleration at the top of the wheel is approximately 1.882 m/s^2.
To determine the direction, we can use the arctan function:
θ = arctan(at/ac)
θ = arctan(-1.8656 m/s^2 / 0.2916 m/s^2)
θ ≈ -83.97 degrees
The direction of the passenger's acceleration at the top of the wheel is approximately 83.97 degrees below the horizontal axis (or clockwise from the positive x-axis).
To find the magnitude and direction of the passenger's acceleration at the top of the Ferris wheel, we need to consider the radial acceleration and tangential acceleration.
First, let's find the radial acceleration. At the top of the Ferris wheel, the radial acceleration can be calculated using the formula:
a_r = r * ω^2
where a_r is the radial acceleration, r is the radius of the Ferris wheel, and ω is the angular velocity.
Substituting the given values, we have:
a_r = 8.8 m * (2π/34.6 s)^2 ≈ 3.15 m/s^2
Next, let's find the tangential acceleration using the formula:
a_t = r * α
where a_t is the tangential acceleration and α is the angular acceleration.
Given that the rate of deceleration is 0.212 rad/s^2, we can use the negative value since it represents deceleration:
α = -0.212 rad/s^2
Substituting the values, we have:
a_t = 8.8 m * (-0.212 rad/s^2) ≈ -1.86 m/s^2
Now, we can find the net acceleration by using vector addition. Since the radial acceleration and tangential acceleration are at right angles to each other, we can use the Pythagorean theorem:
a = √(a_r^2 + a_t^2)
Substituting the values, we have:
a = √((3.15 m/s^2)^2 + (-1.86 m/s^2)^2) ≈ 3.69 m/s^2
Therefore, the magnitude of the passenger's acceleration at the top of the Ferris wheel is approximately 3.69 m/s^2.
To determine the direction, we can use the arctan function to find the angle:
θ = arctan(a_t/a_r)
Substituting the values, we have:
θ = arctan((-1.86 m/s^2) / (3.15 m/s^2)) ≈ -29.76 degrees
Since the result is negative, it means that the acceleration is directed in the opposite direction of the radial acceleration. Therefore, the direction of the passenger's acceleration is approximately 180 - 29.76 degrees, which is around 150.24 degrees in the clockwise direction from the vertical.