A clever design of a bottle holder (a plank with a hole) is shown in the figure. Note that the holder is not fixed to the support; it stands freely, but only when the bottle is in. Assume that the mass of the bottle is 1 kg. The bottle in its rest position is slightly tipped up (α = 15°). Assuming the mass of the holder to be negligible and l = 24 cm, and the angle θ = 53°, then the distance of COM of the bottle from the hole (x in cm) so that the bottle and the holder can stand together as shown is 5n. (Take cos 15° = 24/25.) Find n = ?
Physics · Mixed Chapter
JEE Mains

Question

A clever design of a bottle holder (a plank with a hole) is shown in the figure. Note that the holder is not fixed to the support; it stands freely, but only when the bottle is in. Assume that the mass of the bottle is 1 kg. The bottle in its rest position is slightly tipped up (α = 15°). Assuming the mass of the holder to be negligible and l = 24 cm, and the angle θ = 53°, then the distance of COM of the bottle from the hole (x in cm) so that the bottle and the holder can stand together as shown is 5n. (Take cos 15° = 24/25.) Find n = ?

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The holder is massless, so for balance the bottle's centre of mass must lie vertically above the point of floor support. Step 2: The support hole is a distance l=24 cm along the plank at θ=53°. Its horizontal projection is l cos53°=24(3/5)=14.4 cm. Step 3: The centre-of-mass offset x along the tilted bottle contributes horizontal projection x cos15°. Hence x cos15°=l cos53°, so x=14.4/(24/25)=15 cm. Step 4: Since x=5n cm, n=3.