To find the centripetal force needed to keep the runner on the curve, we can use the formula for centripetal force:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( F_c \) is the centripetal force,
- \( m \) is the mass of the runner (55 kg),
- \( v \) is the speed of the runner (7.2 m/s),
- \( r \) is the radius of the bend (10 m).
Now, let's plug in the values:
- Mass, \( m = 55 \) kg
- Speed, \( v = 7.2 \) m/s
- Radius, \( r = 10 \) m
Now, substituting these values into the equation:
\[ F_c = \frac{55 , \text{kg} \times (7.2 , \text{m/s})^2}{10 , \text{m}} \]
Calculating \( (7.2 , \text{m/s})^2 \):
\[ (7.2)^2 = 51.84 , \text{m}^2/\text{s}^2 \]
Now substituting this back into the centripetal force equation:
\[ F_c = \frac{55 , \text{kg} \times 51.84 , \text{m}^2/\text{s}^2}{10 , \text{m}} \]
Calculating the numerator:
\[ 55 \times 51.84 = 2851.2 , \text{kg} \cdot \text{m}^2/\text{s}^2 \]
Now, dividing by 10 m:
\[ F_c = \frac{2851.2 , \text{kg} \cdot \text{m}^2/\text{s}^2}{10 , \text{m}} = 285.12 , \text{N} \]
Thus, the centripetal force needed to keep the runner on the curve is:
\[ \boxed{285.12 , \text{N}} \]
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