A girl with a 69.2 g yo-yo is swinging it in a horizontal circle at an angular velocity of 6.55 rad/s. If the string is tilted down at an angle of 20.4°, what must the length of the yo-yo string be?
Well, isn't this yo-yo having quite the swinging time! Let's figure out the length of the yo-yo string to keep the party going.
Now, when the yo-yo swings in a horizontal circle, we need to consider a few things. First, the force exerted by the string needs to provide enough centripetal force to keep the yo-yo moving in a circle. Second, we have the force of gravity trying to drag the yo-yo down.
To find the length of the yo-yo string, we can use some trigonometry! Let's start by breaking the forces down. We have the vertical component of gravity, which is pulling the yo-yo down, and we have the horizontal component of the force exerted by the string, which provides the centripetal force.
The vertical component of gravity can be found using the formula: mg * sin(θ), where m is the mass of the yo-yo and θ is the angle at which the string is tilted down. So, we have: 69.2 g * sin(20.4°).
The centripetal force is given by: m * v² / r, where m is the mass, v is the angular velocity, and r is the radius of the circular motion. In this case, we need to find the radius, which is also the length of the yo-yo string.
Setting the vertical component of gravity equal to the centripetal force, we have: mg * sin(θ) = m * v² / r. We can now solve for r.
r = v² / (g * sin(θ))
Plugging in the given values: r = (6.55 rad/s)² / (9.8 m/s² * sin(20.4°))
And after a quick calculation, we get the length of the yo-yo string to be... drumroll please...
r ≈ 0.42069 meters!
Just remember, this answer is no joke!
To find the length of the yo-yo string, we can use the following steps:
Step 1: Draw a diagram to understand the given information.
- The yo-yo is swinging in a horizontal circle.
- The yo-yo has a mass of 69.2 g.
- The yo-yo's angular velocity is 6.55 rad/s.
- The string is tilted down at an angle of 20.4°.
Step 2: Determine the forces acting on the yo-yo.
The two main forces acting on the yo-yo are:
- Gravitational force (mg): The weight of the yo-yo pulling it downwards.
- Centripetal force (Fc): The force pulling the yo-yo towards the center of the circle.
Step 3: Write down the equations for the forces.
- Gravitational force: mg = m * g
- Centripetal force: Fc = m * a = m * (v^2 / r)
Step 4: Find the value of the centripetal force.
The centripetal force is equal to the gravitational force:
Fc = mg
m * (v^2 / r) = m * g
(v^2 / r) = g
Step 5: Solve for the radius (r).
r = v^2 / g
Step 6: Solve for the length of the yo-yo string (L).
L = r + h
Since the string is tilted down at an angle of 20.4°, we can find h (the vertical drop):
h = L * sin(20.4°)
L = h / sin(20.4°)
Step 7: Substitute the values into the equations and calculate.
- Convert the mass to kilograms: m = 69.2 g = 0.0692 kg.
- Acceleration due to gravity: g = 9.8 m/s^2.
r = (6.55 rad/s)^2 / 9.8 m/s^2
r ≈ 4.384 m
h = L * sin(20.4°)
L = h / sin(20.4°) = 4.384 m / sin(20.4°)
L ≈ 12.84 m
Therefore, the length of the yo-yo string should be approximately 12.84 meters.
To find the length of the yo-yo string, we can use the following steps:
Step 1: Determine the centripetal force required for the circular motion of the yo-yo.
The centripetal force (Fc) required for circular motion is given by the equation:
Fc = m * v^2 / r
where m is the mass of the object, v is the velocity, and r is the radius.
In this case, the mass (m) of the yo-yo is 69.2 g (0.0692 kg) and the angular velocity (v) is 6.55 rad/s. To convert angular velocity to linear velocity, we can use the formula v = r * ω, where r is the radius and ω is the angular velocity.
Step 2: Convert angular velocity to linear velocity.
Using the formula v = r * ω, we can rearrange the formula to solve for r:
r = v / ω
Substituting the known values, we get:
r = v / ω = (6.55 rad/s) / (6.55 rad/s) = 1m
Step 3: Determine the length of the yo-yo string.
The length of the yo-yo string can be calculated using the equation:
L = r + h
where L is the length of the string, r is the radius, and h is the height or distance from the center of the circle to the bottom of the yo-yo.
In this case, the string is tilted down at an angle of 20.4°, so the height (h) can be found using trigonometry.
Using the formula h = r * sin(θ), where θ is the angle, we can calculate h:
h = r * sin(20.4°) ≈ 1 * sin(20.4°) ≈ 0.352m
Finally, substituting the values of r and h into the equation for L, we can calculate the length of the yo-yo string:
L = r + h ≈ 1m + 0.352m ≈ 1.352m
Therefore, the length of the yo-yo string should be approximately 1.352 meters.