Answer:
a) To find the acceleration at the bottom of the pipe, we can use the formula for centripetal acceleration: a = v^2 / r where v is the velocity, and r is the radius (half of the diameter) of the pipe. Since the velocity is constant and equal to 5 m/s, and the radius of the pipe is 4 m, the acceleration at the bottom is:
a = (5 m/s)^2 / 4 m a = 6.25 m/s^2
b) To find the force on the bike at an angle of 30° up from the bottom, we need to use the formula for centripetal force: F = m * a where m is the mass of the bike and rider (given as 400 kg) and a is the centripetal acceleration calculated in part (a). The force on the bike at an angle of 30° up from the bottom is:
F = 400 kg * 6.25 m/s^2 * cos(30°) F = 3,464 N
c) To find the minimum velocity at the top for the bike and rider to stay moving in a circle, we can use the same formula for centripetal acceleration and solve for velocity: a = v^2 / r v = sqrt(a * r) where r is the radius of the pipe (again, 4 m) and a is the centripetal acceleration required to keep the bike and rider moving in a circle, which is equal to the acceleration due to gravity at the top of the pipe:
a = g = 9.81 m/s^2 v = sqrt(9.81 m/s^2 * 4 m) v = 6.26 m/s
d) Comparing the minimum velocity calculated in part (c) to the constant speed of 5 m/s given in the question, we can see that the bike and rider do have sufficient velocity to stay moving on a circle at the top of the pipe.