Calculating the tension in the string at the top and bottom of the yo-yo's revolution involves Newton's second law for circular motion, accounting for gravitational forces. At the top, tension is the centripetal force minus the yo-yo's weight, and at the bottom, it is the centripetal force plus the weight. The maximum speed of the yo-yo before the string breaks is calculated using the known breaking tension.
The student has a yo-yo of mass 255 g (0.255 kg) moving in a vertical circle with a speed of 2.5 m/s and a string length of 34.5 cm (0.345 m). To calculate the tension in the string at different positions in the revolution, we will apply Newton's second law for circular motion, considering both the tension and the gravitational force.
At The Top of The Revolution
At the top of the circle, the centripetal force required to keep the yo-yo in circular motion comes from the tension in the string and the component of gravitational force. The total force acting as the centripetal force (Fc) is equal to T (tension) + mg (weight of the yo-yo), since they both point towards the center of the circle. Using the formula Fc = mv2/r:
T + mg = mv2/r
T = mv2/r - mg
Where m = 0.255 kg, g = 9.8 m/s2 (acceleration due to gravity), v = 2.5 m/s, and r = 0.345 m.
At The Bottom of The Revolution
At the bottom of the circle, the tension in the string is counteracted by the weight of the yo-yo. Therefore, the tension is actually more than the centripetal force by the amount of the weight of the yo-yo. The force equation at this point will be:
T - mg = mv2/r
T = mv2/r + mg
Same variables apply as mentioned earlier.
Tension before Yo-Yo String Breaks
To find the maximum speed (vmax) that the yo-yo can have before the tension reaches 22.4 N, we can use the formula at the bottom of the circle as this is where the maximum tension will occur:
22.4 = m * vmax2/r + mg
Solving for vmax will give us the speed at which the string will break.