2. A bicyclist is riding at a tangential speed of 13.2m/s around a circular track. The magnitude of the centripetal force is 377 N, and the combined mass of the bicycle and rider is 86 .5 kg. What is the track's radius?

To find the radius of the circular track, we can use the formula for centripetal force:<br /><br />\[ F_c = \frac{mv^2}{r} \]<br /><br />where:<br />- \( F_c \) is the centripetal force,<br />- \( m \) is the mass of the object,<br />- \( v \) is the tangential speed,<br />- \( r \) is the radius of the circular path.<br /><br />Given:<br />- \( F_c = 377 \, \text{N} \),<br />- \( m = 86.5 \, \text{kg} \),<br />- \( v = 13.2 \, \text{m/s} \).<br /><br />We need to solve for \( r \):<br /><br />\[ r = \frac{mv^2}{F_c} \]<br /><br />Substitute the given values into the equation:<br /><br />\[ r = \frac{86.5 \, \text{kg} \times (13.2 \, \text{m/s})^2}{377 \, \text{N}} \]<br /><br />First, calculate the numerator:<br /><br />\[ 86.5 \, \text{kg} \times (13.2 \, \text{m/s})^2 = 86.5 \times 174.24 = 15110.32 \, \text{kg} \cdot \text{m}^2/\text{s}^2 \]<br /><br />Next, divide by the centripetal force:<br /><br />\[ r = \frac{15110.32 \, \text{kg} \cdot \text{m}^2/\text{s}^2}{377 \, \text{N}} \approx 40.05 \, \text{m} \]<br /><br />Therefore, the radius of the track is approximately \( 40.05 \, \text{m} \).