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Let's analyze the given problem, which involves a frictionless gear train. Here's the breakdown of the situation:

- We have a gear train with four gears.
- The leftmost gear has 12 teeth and is given a torque of 100 N·m.
- The second gear has 48 teeth and is meshed with the first gear.
- The third gear, which is on the same shaft as the second, also has 12 teeth and meshes with the fourth gear, which has 60 teeth.
- The problem asks for the output torque on the 60-teeth gear.

### Step-by-step Solution

1. **Understanding Gear Ratios:**
   The torque transmitted through gears depends on the gear ratio, which is the ratio of the number of teeth on the driving gear to the number of teeth on the driven gear.

   The formula for torque in a gear system is:
   \[
   \text{Torque}_{\text{output}} = \text{Torque}_{\text{input}} \times \left(\frac{\text{Teeth on driven gear}}{\text{Teeth on driving gear}}\right)
   \]

2. **First Gear Pair (12-teeth driving, 48-teeth driven):**
   The 12-teeth gear is driving the 48-teeth gear.
   \[
   \text{Torque}_{\text{on 48-teeth gear}} = 100 \, \text{N·m} \times \left(\frac{48}{12}\right) = 100 \, \text{N·m} \times 4 = 400 \, \text{N·m}
   \]

3. **Second Gear Pair (12-teeth driving, 60-teeth driven):**
   Now, the 12-teeth gear on the same shaft as the 48-teeth gear drives the 60-teeth gear. The torque on the shaft is the same as on the 48-teeth gear, which is 400 N·m. 
   
   The gear ratio here is:
   \[
   \text{Torque}_{\text{on 60-teeth gear}} = 400 \, \text{N·m} \times \left(\frac{60}{12}\right) = 400 \, \text{N·m} \times 5 = 2000 \, \text{N·m}
   \]

### Final Answer:
The output torque on the 60-teeth gear is **2000 N·m**.

Would you like more details or clarifications on any step?

Here are 5 additional questions to extend your understanding:
1. How would the result change if the number of teeth on the output gear was reduced?
2. What is the relationship between torque and rotational speed in gear systems?
3. How does the assumption of "frictionless" gears impact the result?
4. How would you calculate the angular velocity of the output gear?
5. What if the gears had different diameters, but the same number of teeth?

**Tip:** When working with gear ratios, always ensure the driving and driven gears are correctly identified to avoid incorrect ratio calculations.

A frictionless gear train is shown. The leftmost 12-teeth gear is given a torque of 100 N·m. What is the output torque from the 60-teeth gear on the right?

This problem explains how to calculate the output torque in a frictionless gear train system. Starting with a 12-teeth input gear applying a 100 N·m torque, the process explores the torque transmission through a gear train with intermediate gears and provides a detailed step-by-step solution. Suitable for high school physics and engineering students.

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