y ng the expression(1) for the effective acceleration, we have geff=g-a=(g cos Ber-gsin Bee)-(-102e+lee)=(g cos0+102)er, here we have used the fact that g sin 0+10=0, from(2). Thus, we see that the effective gravity points in the radial direction. This explains why when we swing a bottle with fuid inside, the surface of the fuid remains normal to the string Example Plumb and balloon on an aircraft Inside an aircraft there is a plumb su d from the ceiling and a helium filled balloon attached to the foor as shown in the sketch. When the aircraft is fying at a horizontal constant velocity, the tension in the string holding the plumb is 2 N, and the tension in the string holding the balloon is 1 N We want to calculate the angle that the strings form with the vertical and the tension in the strings when: a) the aircraft is at the bottom of a vertical circular loop of radius R, and is flying at a constant speed of b)the aircraft is at the top of a vertical circular loop of radius R, and is flying at a constant speed of U=yg. R2.Using the expression (1) for the effective acceleration, we have, gef f = g − a = (g cos θer − g sin θeθ) − (−l ˙θ 2 er + l ¨θeθ) = (g cos θ + l ˙θ 2 )er , where we have used the fact that g sin θ + l ¨θ = 0, from (2). Thus, we see that the effective gravity points in the radial direction. This explains why when we swing a bottle with fluid inside, the surface of the fluid remains normal to the string. Example Plumb and balloon on an aircraft Inside an aircraft there is a plumb suspended from the ceiling and a helium filled balloon attached to the floor as shown in the sketch. When the aircraft is flying at a horizontal constant velocity, the tension in the string holding the plumb is 2 N, and the tension in the string holding the balloon is 1 N. We want to calculate the angle that the strings form with the vertical and the tension in the strings when: a) the aircraft is at the bottom of a vertical circular loop of radius R, and is flying at a constant speed of v = √ gR. b) the aircraft is at the top of a vertical circular loop of radius R, and is flying at a constant speed of v = √ gR. 4