Cheryl is riding on the edge of a merry-go-round, 2m from the center, which is rotating with an increasing angular speed. Cheryl’s tangential acceleration is 3.0m/s2. At the instant that Cheryl’s linear speed is 4.0m/s, what is Cheryl’s total acceleration?
Express your answer to two significant figures with the appropriate units.
8.5 m/s^2
Well, Cheryl seems to be having quite the ride! With all that spinning, it's understandable that she might be feeling a bit accelerated.
Now, to calculate Cheryl's total acceleration, we need to consider both her tangential acceleration and centripetal acceleration.
The tangential acceleration is given as 3.0 m/s². This represents how fast Cheryl's linear speed is changing.
The centripetal acceleration, on the other hand, is related to the circular motion and can be calculated using the formula ac = v²/r, where ac is the centripetal acceleration, v is the linear speed, and r is the radius.
Given that Cheryl's linear speed is 4.0 m/s and her radius is 2.0 m, we can calculate the centripetal acceleration:
ac = (4.0 m/s)² / 2.0 m = 8.0 m/s²
So Cheryl's centripetal acceleration is 8.0 m/s².
To find the total acceleration, we need to consider both the tangential acceleration and centripetal acceleration. Since they are perpendicular to each other, we can find the total acceleration using the Pythagorean theorem:
atotal = √(at^2 + ac^2)
Plugging in the values we know:
atotal = √((3.0 m/s²)² + (8.0 m/s²)²) ≈ 8.6 m/s²
Therefore, Cheryl's total acceleration is approximately 8.6 m/s². I hope Cheryl is enjoying the thrill of her merry-go-round adventure!
To find Cheryl's total acceleration, we'll first need to calculate her centripetal acceleration and then combine it with her tangential acceleration.
Let's start by calculating the centripetal acceleration using the formula:
ac = ω² * r
Where:
ac is the centripetal acceleration,
ω is the angular speed, and
r is the distance from the center.
Given that Cheryl is riding 2m from the center and her angular speed is increasing, we don't have the specific value for ω. However, we know that linear speed is related to angular speed by the formula:
v = ω * r
Rearranging the formula to solve for ω:
ω = v / r
Substituting the values, we have:
ω = 4.0 m/s / 2 m
ω = 2.0 rad/s
Now, we can calculate the centripetal acceleration:
ac = (2.0 rad/s)² * 2 m
ac = 4.0 m/s²
Next, we need to find the total acceleration by combining the centripetal and tangential accelerations. Since these accelerations act perpendicular to each other, we can use the Pythagorean theorem:
atotal = sqrt(ac² + at²)
Where:
atotal is the total acceleration, and
at is the tangential acceleration.
Substituting the given value of at:
atotal = sqrt((4.0 m/s²)² + (3.0 m/s²)²)
atotal = sqrt(16.0 m²/s⁴ + 9.0 m²/s⁴)
atotal = sqrt(25.0 m²/s⁴)
atotal = 5.0 m/s²
Therefore, Cheryl's total acceleration is 5.0 m/s².