Mary and her younger brother Alex decide to ride the carousel at the State Fair. Mary sits on one of the horses in the outer section at a distance of 2.0 m from the center. Alex decides to play it safe and chooses to sit in the inner section at a distance of 1.1 m from the center. The carousel takes 4.8 s to make each complete revolution. (a) What is Mary's angular speed ωM and tangential speed vM? ωM = rev/s vM = m/s (b) What is Alex's angular speed ωA and tangential speed vA? ωA = rev/s vA = m/s
To find the angular speed ωM and tangential speed vM for Mary, we can use the formula:
ω = θ / t
where ω is the angular speed in rev/s, θ is the angle in radians, and t is the time in seconds.
(a) For Mary:
- The carousel takes 4.8 seconds to make a complete revolution, which is equivalent to 2π radians.
- So, θM = 2π radians and t = 4.8 seconds.
Using the formula, we can calculate ωM:
ωM = θM / t
= 2π / 4.8
≈ 1.31 rev/s
Now, to find the tangential speed vM, we will use the formula:
v = rω
where v is the tangential speed in m/s, r is the radius in meters, and ω is the angular speed in rev/s.
- For Mary, rM = 2.0 m and ωM = 1.31 rev/s.
Using the formula, we can calculate vM:
vM = rM * ωM
= 2.0 * 1.31
≈ 2.62 m/s
Therefore, Mary's angular speed ωM is approximately 1.31 rev/s, and her tangential speed vM is approximately 2.62 m/s.
(b) For Alex:
- The carousel takes the same time (4.8 seconds) to make a complete revolution, which is equivalent to 2π radians.
- So, θA = 2π radians and t = 4.8 seconds.
Using the formula, we can calculate ωA:
ωA = θA / t
= 2π / 4.8
≈ 1.31 rev/s
Now, to find the tangential speed vA, we use the same formula as before:
vA = rA * ωA
- For Alex, rA = 1.1 m and ωA = 1.31 rev/s.
Using the formula, we can calculate vA:
vA = rA * ωA
= 1.1 * 1.31
≈ 1.44 m/s
Therefore, Alex's angular speed ωA is approximately 1.31 rev/s, and his tangential speed vA is approximately 1.44 m/s.
To find the angular speed (ω) and tangential speed (v) for Mary and Alex, we can use the following formulas:
Angular speed (ω) = 2π / Time taken for one revolution
Tangential speed (v) = Radius × Angular speed
(a) Mary's angular speed (ωM):
The time taken for one revolution is given as 4.8 s. So, we can use the formula:
ωM = 2π / 4.8
To calculate ωM, we divide 2π (approximately 6.28) by 4.8:
ωM = 6.28 / 4.8 ≈ 1.307 rev/s
Mary's angular speed (ωM) is approximately 1.307 revolutions per second.
Mary's tangential speed (vM):
The radius of Mary's position on the carousel is given as 2.0 m. Thus, we can use the formula:
vM = Radius × ωM
Substituting the values, we get:
vM = 2.0 × 1.307 ≈ 2.614 m/s
Mary's tangential speed (vM) is approximately 2.614 m/s.
(b) Alex's angular speed (ωA):
Like Mary, Alex's angular speed can be found using the formula:
ωA = 2π / 4.8
We divide 2π by 4.8:
ωA = 6.28 / 4.8 ≈ 1.307 rev/s
Alex's angular speed (ωA) is approximately 1.307 revolutions per second.
Alex's tangential speed (vA):
The radius for Alex's position is given as 1.1 m. Using the formula:
vA = Radius × ωA
Substituting the given values, we find:
vA = 1.1 × 1.307 ≈ 1.438 m/s
Alex's tangential speed (vA) is approximately 1.438 m/s.
Therefore, the answers are:
(a) Mary's angular speed ωM = 1.307 rev/s, and tangential speed vM = 2.614 m/s.
(b) Alex's angular speed ωA = 1.307 rev/s, and tangential speed vA = 1.438 m/s.
a) w=2PI/4.8, vt=w*r
b) w=2PI/4.8, vt=w*r