find the minimum speed for the roller coaster riding along the inner side of a loop with a radius r and an acceleration of 1.5 times earth's little g for each of the following for each of the following rollercoaster constructions:
a.the radius of curvature at the top of the loop is 15 m.
b.the radius of curvature at the top of the loop is 25 m.
To find the minimum speed for a roller coaster riding along the inner side of a loop, we can use the concept of centripetal force.
a. For the first roller coaster construction where the radius of curvature at the top of the loop is 15 m, let's calculate the minimum speed.
Step 1: Find the acceleration at the top of the loop. It is given as 1.5 times the acceleration due to gravity (g).
Acceleration (a) = 1.5 * g
Step 2: Calculate the net force at the top of the loop. The net force is the difference between the gravitational force (mg) and the centripetal force (mv^2/r). At the top of the loop, the net force should be pointing towards the center of the loop.
Net Force (F_net) = mg - (mv^2 / r)
Step 3: Equate the net force to the centripetal force (mv^2 / r) and solve for velocity (v).
mv^2 / r = mg - (mv^2 / r)
Simplify the equation:
mv^2 / r + mv^2 / r = mg
2mv^2 / r = mg
v^2 = (rg) / 2
v = sqrt((rg) / 2)
Substitute the value of g (gravitational acceleration) and r (radius of curvature at the top of the loop) to get the minimum speed for the roller coaster.
v = sqrt((1.5g * 15) / 2)
b. For the second roller coaster construction where the radius of curvature at the top of the loop is 25 m, we can follow the same steps as above:
v = sqrt((1.5g * 25) / 2)
Remember to use the appropriate value for the acceleration due to gravity (g), which is approximately 9.8 m/s^2. Calculate the minimum speed for each roller coaster by substituting the values and simplify the expression using a calculator or by hand.