Centripetal Force 1. A 2.10 m rope attaches a tire to an overhanging tree limb . A girl swinging on the tire has a tangential speed of 2.50m/s. If the magnitude of the centripetal force is 88 .0 N, what is the girl's mass?

To find the girl's mass, we can use the formula for centripetal force:<br /><br />\[ F_c = \frac{m \cdot v^2}{r} \]<br /><br />where:<br />- \( F_c \) is the centripetal force,<br />- \( m \) is the mass of the girl,<br />- \( v \) is the tangential speed,<br />- \( r \) is the radius of the circular path.<br /><br />Given:<br />- \( F_c = 88.0 \, \text{N} \)<br />- \( v = 2.50 \, \text{m/s} \)<br />- \( r = 2.10 \, \text{m} \)<br /><br />We need to solve for \( m \):<br /><br />\[ m = \frac{F_c \cdot r}{v^2} \]<br /><br />Substitute the given values into the equation:<br /><br />\[ m = \frac{88.0 \, \text{N} \cdot 2.10 \, \text{m}}{(2.50 \, \text{m/s})^2} \]<br /><br />Calculate the denominator:<br /><br />\[ (2.50 \, \text{m/s})^2 = 6.25 \, \text{m}^2/\text{s}^2 \]<br /><br />Now, calculate the mass:<br /><br />\[ m = \frac{88.0 \, \text{N} \cdot 2.10 \, \text{m}}{6.25 \, \text{m}^2/\text{s}^2} \]<br /><br />\[ m = \frac{184.8 \, \text{N} \cdot \text{m}}{6.25 \, \text{m}^2/\text{s}^2} \]<br /><br />\[ m = 29.376 \, \text{kg} \]<br /><br />Therefore, the girl's mass is approximately \( 29.38 \, \text{kg} \).