A sprinter of mass 63.7 kg runs at a constant speed of 10.2 m/s in a straight line. Find the extra force the sprinter must exert on the ground to round a curve of radius 10.8 m.
To find the extra force exerted by the sprinter, we need to consider the centripetal force required to keep the sprinter moving in a curved path. The centripetal force is directed towards the center of the circle and is equal to the mass multiplied by the square of the velocity, divided by the radius of the curve.
First, let's calculate the centripetal force:
Centripetal Force = (Mass x Velocity^2) / Radius
Given:
Mass (m) = 63.7 kg
Velocity (v) = 10.2 m/s
Radius (r) = 10.8 m
Plugging in the values:
Centripetal Force = (63.7 kg x (10.2 m/s)^2) / 10.8 m
To simplify the calculation, let's break it down into steps.
Step 1: Square the velocity
Velocity^2 = (10.2 m/s)^2
Step 2: Calculate the centripetal force
Centripetal Force = (63.7 kg x (105.84 m^2/s^2)) / 10.8 m
Step 3: Simplify the formula
Centripetal Force = (6827.9088 kg⋅m^2/s^2) / 10.8 m
Step 4: Divide to find the extra force
Centripetal Force = 633.1437 kg⋅m/s^2
Therefore, the sprinter must exert an extra force of 633.1437 N on the ground to round the curve.