To determine the force exerted by the pavement on the tires to keep a vehicle moving in a circular path, we need to calculate the centripetal force. The formula for centripetal force [tex]\( F_c \)[/tex] is given by:
[tex]\[ F_c = \frac{m \cdot v^2}{r} \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the vehicle.
- [tex]\( v \)[/tex] is the velocity of the vehicle.
- [tex]\( r \)[/tex] is the radius of the circular path.
Given data:
- The mass of the vehicle [tex]\( m = 341,000 \)[/tex] kg
- The velocity [tex]\( v = 10 \)[/tex] m/s
- The radius [tex]\( r = 10 \)[/tex] m
Now, let's plug these values into the formula:
[tex]\[ F_c = \frac{341,000 \text{ kg} \cdot (10 \text{ m/s})^2}{10 \text{ m}} \][/tex]
First, calculate the velocity squared:
[tex]\[ v^2 = (10 \text{ m/s})^2 = 100 \text{ m}^2/\text{s}^2 \][/tex]
Then, multiply the mass by the velocity squared:
[tex]\[ 341,000 \text{ kg} \cdot 100 \text{ m}^2/\text{s}^2 = 34,100,000 \text{ kg} \cdot \text{m}^2/\text{s}^2 = 34,100,000 \text{ N} \cdot \text{m} \][/tex]
Finally, divide this product by the radius:
[tex]\[ F_c = \frac{34,100,000 \text{ N} \cdot \text{m}}{10 \text{ m}} = 3,410,000 \text{ N} \][/tex]
Hence, the force exerted by the pavement on the tires to keep the vehicle in the circular path is [tex]\( 3,410,000 \)[/tex] N. Since the calculated force is [tex]\( 3,410,000 \text{ N} = 3.41 \times 10^6 \text{ N} \)[/tex], let's compare it with the given options.
None of the options A to E match this exact value. However, it seems there might be a typo in the question options. Given the calculation, the correct answer should be:
[tex]\[ \boxed{3,410,000 \text{ N}} \][/tex]
