1. **State the problem:** We need to find the normal reaction force acting on a roller coaster at the top of a vertical circular loop. 2. **Given data:** - Speed at top of loop, $v = 21$ m/s - Radius of curvature, $r = 85$ m - Mass of roller coaster and riders, $m = 1550$ kg 3. **Relevant physics and formula:** At the top of the loop, the forces acting on the roller coaster are: - Gravitational force downward: $mg$ - Normal reaction force from the track: $N$ (also downward at the top since coaster is upside down) The centripetal force required to keep the coaster moving in a circle is provided by the sum of these forces: $$ F_c = m \frac{v^2}{r} = mg + N $$ Rearranging to solve for $N$: $$ N = m \frac{v^2}{r} - mg $$ 4. **Calculate each term:** $$ m \frac{v^2}{r} = 1550 \times \frac{21^2}{85} = 1550 \times \frac{441}{85} $$ Calculate the fraction: $$ \frac{441}{85} \approx 5.1882 $$ So: $$ 1550 \times 5.1882 = 8041.71 \text{ N} $$ Calculate gravitational force: $$ mg = 1550 \times 9.8 = 15190 \text{ N} $$ 5. **Calculate normal force:** $$ N = 8041.71 - 15190 = -7148.29 \text{ N} $$ 6. **Interpretation:** The negative sign means the normal force acts opposite to the assumed direction (upward), so the track is pulling on the coaster with a force of 7148.29 N upward. **Final answer:** $$ \boxed{N = -7148.29 \text{ N}} $$ This means the normal force magnitude is 7148.29 N directed upward at the top of the loop.