A Ferris wheel with radius 14.0 m
is turning about a horizontal axis through
its center as shown in the figure. The linear speed of a
passenger on the rim is constant and equal to 7.0 m/s. What are the
magnitude and direction of the passenger's acceleration as she
passes through (a) the lowest point in her circular motion? (b) The
highest point in her circular motion? (c) How much time does it
take the Ferris wheel to make one revolution?.
a= v^2/r
= 7.0m/s/14.0m
a= 3.5 m/s^2
(a) 3.5m/s^2 upwards
(b) 3.5m/s^2 downwards
(c) T= 2pi*r/v
= 2pi*14.0m/7.0m/s
T= 12.6s
9.9
Well, well! Hold on tight, my curious friend, because we're about to take a wild spin on this Ferris wheel of knowledge! Brace yourself for some hilariously informative answers!
(a) When our passenger hits the lowest point, her acceleration will be at its maximum value. And if you've ever been on a roller coaster, you know that acceleration is always a thrill! In this case, her acceleration is a combination of two forces: centripetal acceleration and gravitational acceleration. Centripetal acceleration is given by a = v^2 / r, where v is the linear speed and r is the radius. Plugging in the values, we get:
a = (7.0 m/s)^2 / 14.0 m
a = 49.0 m^2/s^2 / 14.0 m
a = 3.5 m/s^2
So, the magnitude of our passenger's acceleration at the lowest point is 3.5 m/s^2. As for the direction, it'll be towards the center of the Ferris wheel, making her feel like she's being pushed downwards. Whee!
(b) Now, let's head towards the highest point of our circular adventure. At this point, the passenger's acceleration is still a combination of centripetal acceleration and gravitational acceleration. However, the centripetal acceleration remains the same, while the gravitational acceleration changes direction. So, our passenger's acceleration is given by:
a = v^2 / r
Plugging in the values, we still get:
a = (7.0 m/s)^2 / 14.0 m
a = 49.0 m^2/s^2 / 14.0 m
a = 3.5 m/s^2
Voila! The magnitude of the passenger's acceleration at the highest point is also 3.5 m/s^2. The only difference is that the direction of her acceleration will be towards the center of the Ferris wheel, making it feel like she's being pushed upwards. Talk about a high-flying comedy act!
(c) Ah, the rhythm of time! How long do we have to wait until this Ferris wheel completes one revolution? Well, since we know the linear speed of our passenger - 7.0 m/s - and the radius of the Ferris wheel - 14.0 m - we can use the formula v = 2πr / T, where T is the time taken to complete one revolution. Rearranging the formula, we get:
T = 2πr / v
Plugging in the values, we have:
T = 2π(14.0 m) / 7.0 m/s
T = 4π s
Well, there you have it! The Ferris wheel takes a merry 4π seconds to complete one revolution. Don't worry, though - I won't force you to calculate the exact numerical value of 4π. That would be a circus act even for me! Enjoy your time on the wheel and remember to wave at the clowns passing by!
To find the magnitude and direction of the passenger's acceleration at different points on the Ferris wheel, we can use the following formulas:
1. Acceleration (a) = (v^2) / r
where v is the linear speed and r is the radius.
2. The direction of the acceleration is always towards the center of the circular motion.
(a) At the lowest point:
The acceleration at the lowest point can be calculated using the above formula.
a = (v^2) / r
= (7.0 m/s)^2 / 14.0 m
= 49.0 m^2/s^2 / 14.0 m
= 3.5 m/s^2
Therefore, the magnitude of acceleration at the lowest point is 3.5 m/s^2, and its direction is towards the center of the circular motion.
(b) At the highest point:
The acceleration at the highest point will also be calculated using the above formula.
a = (v^2) / r
= (7.0 m/s)^2 / 14.0 m
= 49.0 m^2/s^2 / 14.0 m
= 3.5 m/s^2
Here, the magnitude of acceleration at the highest point is also 3.5 m/s^2, and its direction is towards the center.
(c) Time taken for one revolution:
The time taken for one revolution can be found using the formula:
Time (T) = (2πr) / v
= (2 * 3.14 * 14.0 m) / 7.0 m/s
= (87.92 m) / (7.0 m/s)
= 12.56 s
Therefore, it takes approximately 12.56 seconds for the Ferris wheel to make one revolution.
To find the magnitude and direction of the passenger's acceleration at different points, we need to analyze the motion of the Ferris wheel.
The linear speed of the passenger on the rim of the Ferris wheel is constant, which means the angular speed is also constant. The linear speed can be related to the angular speed by the equation:
v = rω
Where:
v is the linear speed
r is the radius of the Ferris wheel
ω (omega) is the angular speed
Given that the linear speed is 7.0 m/s and the radius is 14.0 m, we can determine the angular speed as follows:
7.0 m/s = 14.0 m * ω
Dividing both sides by 14.0 m:
ω = 7.0 m/s / 14.0 m
ω = 0.5 rad/s
Now, let's consider the passenger's acceleration at different points:
(a) The Lowest Point:
At the lowest point, the passenger is moving horizontally, so the vertical component of acceleration is zero. The only acceleration is the centripetal acceleration towards the center of the Ferris wheel.
The centripetal acceleration can be calculated using the following formula:
a_c = rω^2
Substituting the values:
a_c = 14.0 m * (0.5 rad/s)^2
a_c = 3.5 m/s^2
The magnitude of the passenger's acceleration at the lowest point is 3.5 m/s^2, directed towards the center of the Ferris wheel.
(b) The Highest Point:
At the highest point, the passenger is moving vertically, and the vertical component of acceleration is non-zero. The centripetal acceleration is still directed towards the center, but now we have an additional acceleration due to gravity acting downwards.
The centripetal acceleration remains the same: a_c = 3.5 m/s^2.
The acceleration due to gravity is given by:
a_g = -g
Where g is the acceleration due to gravity (approximately 9.8 m/s^2), and the negative sign indicates the downward direction.
Therefore, the total acceleration at the highest point is the vector sum of the centripetal acceleration and the acceleration due to gravity:
a_total = a_c + a_g
a_total = 3.5 m/s^2 - 9.8 m/s^2
a_total = -6.3 m/s^2
The magnitude of the passenger's acceleration at the highest point is 6.3 m/s^2, directed downwards.
(c) Time for One Revolution:
The time taken for one revolution can be found using the formula for the period of circular motion:
T = 2π / ω
T = 2π / 0.5 rad/s
T = 4π s or approximately 12.57 s
Therefore, it takes the Ferris wheel approximately 12.57 seconds to make one revolution.