2. A child throws his yoyo horizontally about his head rather than using it properly. The yoyo has a mass of 0.200 kg and is attached to a string 0.800 meters long.
a. If the yoyo makes a complete revolution each second, what tension must exist in the string?
b. If the tension in the string is doubled, what is the speed of the yoyo?
Thanks in advance
a. To find the tension in the string, we can use the centripetal force formula. The centripetal force is given by the formula: F = (m * v^2) / r, where F is the centripetal force, m is the mass of the yoyo, v is the velocity, and r is the radius of the circular path (which is equal to the length of the string).
In this case, we know the mass of the yoyo is 0.200 kg, and the radius of the circular path is 0.800 meters. The yoyo makes a complete revolution each second, which means its velocity is equal to the circumference of the circle divided by the time taken for one revolution. So, the velocity is v = (2πr) / t, where t is the time taken for one revolution.
Substituting the values into the formula, the tension in the string can be found using the equation F = (m * ((2πr) / t)^2) / r.
b. If the tension in the string is doubled, we can use the same formula as before to find the new velocity of the yoyo. However, this time we will consider the new tension as 2 times the original tension. So, the new centripetal force is given by: 2F = (m * v'^2) / r, where v' is the new velocity.
Rearranging the formula, we can find v'^2 = (2F * r) / m. Taking the square root of both sides, we can calculate the new velocity of the yoyo by finding the square root of ((2 * the new tension * the radius) divided by the mass of the yoyo).
a. To find the tension in the string, we can use the centripetal force equation:
F = (m * v^2) / r
Where:
F = tension in the string
m = mass of the yoyo = 0.200 kg
v = velocity of the yoyo (circumference of the circular path / time for one revolution)
r = radius of the circular path = length of the string = 0.800 m
First, let's calculate the velocity:
Since the yoyo makes one complete revolution each second, the time for one revolution is 1 second. Therefore, the velocity can be calculated by dividing the circumference of the circular path by the time for one revolution:
Circumference = 2 * π * r
= 2 * π * 0.800 m
≈ 5.026 m
velocity = Circumference / time
= 5.026 m / 1 s
≈ 5.026 m/s
Now, substitute the values into the centripetal force equation:
F = (m * v^2) / r
= (0.200 kg * (5.026 m/s)^2) / 0.800 m
≈ 1.575 N
Therefore, the tension in the string must be approximately 1.575 N.
b. If the tension in the string is doubled, we can use the same centripetal force equation to find the new velocity.
Let's assume the new tension in the string is 2 times the original tension (2 * 1.575 N):
F = (m * v^2) / r
2 * F = (m * v^2) / r
Since the mass of the yoyo and the radius of the circular path remain the same, we can rewrite the equation:
2 * 1.575 N = (0.200 kg * v^2) / 0.800 m
Simplifying the equation:
V^2 = (2 * 1.575 N * 0.800 m) / 0.200 kg
= 12.6 m/s
Taking the square root of both sides to solve for v:
v = √(12.6 m/s)
≈ 3.55 m/s
Therefore, if the tension in the string is doubled, the speed of the yoyo will be approximately 3.55 m/s.