What is the magnitude of the force F on the particle?

Steps to Solve

1. Identify the variables We have the following given values:

• Mass of the particle, $m = 2.4 , \text{kg}$
• Velocity of the particle, $v = 6.0 , \text{m/s}$
• Radius of the circular path, $r = 1.6 , \text{m}$

2. Use the centripetal force formula The formula for centripetal force is given by:

$$ F = \frac{mv^2}{r} $$

Substituting the values into this formula:

$$ F = \frac{(2.4 , \text{kg}) (6.0 , \text{m/s})^2}{1.6 , \text{m}} $$

3. Calculate the square of the velocity First, calculate $v^2$:

$$ v^2 = (6.0 , \text{m/s})^2 = 36.0 , \text{m}^2/\text{s}^2 $$

4. Substitute and simplify Now substitute $v^2$ back into the formula:

$$ F = \frac{(2.4 , \text{kg}) (36.0 , \text{m}^2/\text{s}^2)}{1.6 , \text{m}} $$

5. Calculate the numerator Calculate the numerator:

$$ 2.4 , \text{kg} \times 36.0 , \text{m}^2/\text{s}^2 = 86.4 , \text{kg} \cdot \text{m}^2/\text{s}^2 $$

6. Divide by the radius Now calculate the force:

$$ F = \frac{86.4 , \text{kg} \cdot \text{m}^2/\text{s}^2}{1.6 , \text{m}} = 54.0 , \text{N} $$

7. Final result Thus, the magnitude of the force on the particle is (54 , \text{N}).