A girl with a 36.3 g yo-yo is swinging it in a horizontal circle at an angular velocity of 7.68 rad/s. If the string is tilted down at an angle of 19.7°, what must the length of the yo-yo string be?
Question
A girl with a 36.3 g yo-yo is swinging it in a horizontal circle at an angular velocity of 7.68 rad/s. If the string is tilted down at an angle of 19.7°, what must the length of the yo-yo string be?
Solution
To solve this problem, we need to use the formula for centripetal force and the formula for the force of gravity.
The centripetal force (Fc) is given by the formula Fc = m * ω^2 * r, where m is the mass, ω is the angular velocity, and r is the radius (which is the same as the length of the yo-yo string in this case).
The force of gravity (Fg) is given by the formula Fg = m * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since the string is tilted, the centripetal force is not equal to the force of gravity, but is equal to the force of gravity times the cosine of the angle of the tilt. So we have the equation Fc = Fg * cos(θ).
Substituting the formulas for Fc and Fg into this equation gives us m * ω^2 * r = m * g * cos(θ).
We can solve this equation for r (the length of the yo-yo string) to get r = (g * cos(θ)) / ω^2.
Substituting the given values into this equation gives us r = (9.8 m/s^2 * cos(19.7°)) / (7.68 rad/s)^2.
Calculating this gives us r ≈ 0.33 m, or 33 cm. So the length of the yo-yo string must be approximately 33 cm.
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