Problem 5. Carousel that’s turning A carousel has two lanes of horses: an outer lane of radius 5.36 m and an iner lane of radius 3.78 m. When a ride starts, the carousel immediately starts rotating with some initial angular speed wo, and undergoes a constant angular acceleration a until it reaches its maximum angular speed, after which it rotates constantly at this angular speed. The carousel completes its first revolution in 23.0 s, then completes its second revolution after an additional 21.0s. At its maximum speed, the carousel completes one revolution every 14.0 s. During the ride, the horses also move up and down, and their vertical motion is independent of the angular speed of the carousel. The vertical displacement of each horse follows that of simple harmonic motion, with one up-and-down cycle lasting 3.50 s, and the height difference between each horse's highest point and lowest point being 20.3 cm. All horses start at their lowest point at the beginning of the ride. (a) Find angular acceleration a and initial angular speed wo of the carousel, and find the time at which the carousel reaches its maximum angular speed. (b) At t = 1.00 min, t = 2.00 min, and t = 3.00 min after the beginning of the ride, draw the velocity and acceleration vectors of a horse in the outer lane and a horse in the inner lane, with clearly labeled components. (If needed for clarity, you may draw both a top view and a side view.)
Problem 5. Carousel that’s turning A carousel has two lanes of horses: an outer lane of radius 5.36 m and an iner lane of radius 3.78 m. When a ride starts, the carousel immediately starts rotating with some initial angular speed wo, and undergoes a constant angular acceleration a until it reaches its maximum angular speed, after which it rotates constantly at this angular speed. The carousel completes its first revolution in 23.0 s, then completes its second revolution after an additional 21.0s. At its maximum speed, the carousel completes one revolution every 14.0 s. During the ride, the horses also move up and down, and their vertical motion is independent of the angular speed of the carousel. The vertical displacement of each horse follows that of simple harmonic motion, with one up-and-down cycle lasting 3.50 s, and the height difference between each horse's highest point and lowest point being 20.3 cm. All horses start at their lowest point at the beginning of the ride. (a) Find angular acceleration a and initial angular speed wo of the carousel, and find the time at which the carousel reaches its maximum angular speed. (b) At t = 1.00 min, t = 2.00 min, and t = 3.00 min after the beginning of the ride, draw the velocity and acceleration vectors of a horse in the outer lane and a horse in the inner lane, with clearly labeled components. (If needed for clarity, you may draw both a top view and a side view.)
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Transcribed Image Text:Problem 5. Carousel that’s turning
A carousel has two lanes of horses: an outer lane of radius 5.36 m and an inner lane of radius 3.78 m.
When a ride starts, the carousel immediately starts rotating with some initial angular speed wo,
and undergoes a constant angular acceleration a until it reaches its maximum angular speed, after
which it rotates constantly at this angular speed. The carousel completes its first revolution in 23.0 s,
then completes its second revolution after an additional 21.0s. At its maximum speed, the carousel
completes one revolution every 14.0 s.
During the ride, the horses also move up and down, and their vertical motion is independent of
the angular speed of the carousel. The vertical displacement of each horse follows that of simple
harmonic motion, with one up-and-down cycle lasting 3.50 s, and the height difference between each
horse's highest point and lowest point being 20.3 cm. All horses start at their lowest point at the
beginning of the ride.
(a) Find angular acceleration a and initial angular speed wo of the carousel, and find the time at
which the carousel reaches its maximum angular speed.
(b) At t = 1.00 min, t = 2.00 min, and t = 3.00 min after the beginning of the ride, draw the velocity
and acceleration vectors of a horse in the outer lane and a horse in the inner lane, with clearly
labeled components. (If needed for clarity, you may draw both a top view and a side view.)
Transcribed Image Text:Problem 7. Why did summer go so quickly?
Scientists are studying a planet in a faraway galaxy that has the potential to sustain life. It has the
same mass 5.97 × 1024 kg as our Earth, and orbits a star with the same mass 1.99 × 1030 kg as our
sun. However, its orbit has appreciable eccentricity, so that the distance between this planet and its
star varies between 1.20 x 1011 m and 1.80 x 1011 m during its perihelion and aphelion, respectively.
(a) Find the semimajor axis and the eccentricity of this planet. What is its period of orbit in Earth
days?
(b) Find the speeds (both linear and angular) of this planet at its perihelion and aphelion.
(c) Unlike our Earth, whose seasonal cycle is a result of the tilt of Earth's axis, this planet's rotation
axis has no tilt (i.e. it is perpendicular to the plane of its orbit), and this planet's seasons depend
on its distance from its star. Suppose that the season of summer on this planet occurs when
its distance from its star is less than or equal to 1.40 × 101' m. How many Earth days does the
season of summer last on this planet? Hint: You may find the following integral useful:
Va? – x² dx =
x² + a² sin°
+C
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