Answer :
Final answer:
The yoyo must start with an angular velocity of approximately [tex]\( 1.86 \, \text{rad/s} \)[/tex] to climb the string.
Explanation:
The angular velocity [tex](\(\omega\))[/tex] required for the yoyo to climb the string can be calculated using the conservation of energy principle. The gravitational potential energy lost by the yoyo is converted into rotational kinetic energy. The formula for the angular velocity is given by:
[tex]\[ \omega = \sqrt{\frac{2gh}{R}} \][/tex]
where [tex]\( g \)[/tex] is the acceleration due to gravity, [tex]\( h \)[/tex] is the height climbed, and [tex]\( R \)[/tex] is the radius of rotation.
The height climbed by the yoyo is the length of the string [tex](\( 84 \, \text{cm} \))[/tex]. The radius of rotation is the radius of the yoyo, given in meters as [tex]\( 2.6 \, \text{cm} \times 0.01 \, \text{m/cm} \).[/tex]
Substituting these values into the formula:
[tex]\[ \omega = \sqrt{\frac{2 \times 9.8 \times 0.84}{0.026}} \approx 1.86 \, \text{rad/s} \][/tex]
Therefore, the yoyo must start with an angular velocity of approximately [tex]\( 1.86 \, \text{rad/s} \)[/tex] to climb the string.
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