Question

A system of gears and belts is shown below. The diameters of the gears M, N, P, Q, R and S are 3, 1, 2, 1, 3 and 2 meters respectively. If the gear M is rotating at 100 rpm, what is the rpm of gear S?

Answer 1

Correct Answer: 75

To find the rpm of gear S, we must understand how gear diameters affect their rotational speeds. A smaller gear connected directly to a larger gear rotates faster, and vice versa. We start by examining the sequence of gears and belts:

• Gear M and N are directly connected. Since their diameters are 3 and 1 meters respectively, they have a gear ratio of 3:1.
• Gear N and P are linked via a belt. Since both have the same effective diameter (1 meter), the belt ensures they rotate at the same speed.
• Gear P and Q are directly connected with diameters of 2 and 1 meters respectively, so they have a gear ratio of 2:1.
• Gear Q and R are linked via a belt, rotating at the same rate since their effective diameters are 1 meter each.
• Finally, gears R and S are connected and have diameters of 3 and 2 meters respectively, giving a gear ratio of 3:2.

We'll calculate the rotations stepwise, considering each interaction:

1. Gear M rotates at 100 rpm. Thus, gear N rotates at \(100 \times \frac{3}{1} = 300\) rpm.
2. Gear N is linked via belt to gear P, so gear P also rotates at 300 rpm.
3. Gear P rotates gear Q at a ratio of 2:1, so gear Q spins at \(300 \times \frac{2}{1} = 600\) rpm.
4. Gear Q is linked via belt to gear R, so gear R also rotates at 600 rpm.
5. Finally, gear R rotates gear S at a ratio of 3:2, thus gear S spins at \(600 \times \frac{3}{2} = 900\) rpm.

The computed rpm of gear S is 900, which unfortunately falls outside the specified range of 75 to 75 rpm, indicating either a problem misinterpretation or range error. Please double-check the gear system setup for any missing constraints or specifics on belt connections that might alter our calculations. If the given range is correct, the setup likely requires additional physical constraints not described herein.