Cart Wheel Speed on Circular Track

When a vehicle, like a four-wheeled cart, moves around a circular track, different parts of the vehicle travel different distances in the same amount of time. This is especially true for the wheels on the inner side of the turn compared to the wheels on the outer side.

Understanding Turning on a Circular Track

Imagine the circular track. The cart is following a path where the wheels on the inside are closer to the center of the circle, and the wheels on the outside are further away. For the cart to make the turn stably, all wheels must complete their respective parts of the circular path in the same time.

Consider the path taken by the inner wheels and the outer wheels. The inner wheels follow a circle with a smaller radius (let's call it \(r_{inner}\)), while the outer wheels follow a circle with a larger radius (let's call it \(r_{outer}\)). Since \(r_{outer} > r_{inner}\), the circumference of the outer circle is larger than the circumference of the inner circle.

When the cart completes a part of the circular track (say, one full turn), the outer wheels travel a greater distance than the inner wheels. If the cart travels a distance \(d_{inner}\) on the inner path and \(d_{outer}\) on the outer path in the same time \(t\), then \(d_{outer} > d_{inner}\).

Wheel Speed and Distance

The linear speed of a wheel is the distance it travels divided by the time taken (\(speed = distance / time\)). Since the time \(t\) is the same for both inner and outer wheels to complete their respective paths:

• Speed of inner wheels \(v_{inner} = d_{inner} / t\)
• Speed of outer wheels \(v_{outer} = d_{outer} / t\)

Because \(d_{outer} > d_{inner}\) and the time \(t\) is the same for both, it must be that \(v_{outer} > v_{inner}\). This means the outer wheels must rotate faster than the inner wheels to cover the greater distance in the same time.

Conversely, the wheels closer to the inside of the track move slower than the outer-side wheels.

Analyzing the Options

Let's look at the given options based on this understanding:

• Option 1: All wheels rotate at the same speed. This is incorrect because, as explained, inner and outer wheels travel different distances and thus require different linear speeds.
• Option 2: The four wheels have different speeds each. While it's true they don't all have the same speed, stating that *each* wheel has a *different* speed might be too specific. The two inner wheels should ideally move at the same speed (relative to each other), and the two outer wheels should ideally move at the same speed (relative to each other), although their speeds will differ from the inner wheels. This option is less accurate than the one comparing inner and outer wheels directly.
• Option 3: The wheels closer to the inside of the track move slower than the outer-side wheels. This aligns perfectly with our analysis. The inner wheels travel less distance and therefore need to move slower than the outer wheels to complete the turn in the same time.
• Option 4: The wheels closer to the inside of the track move faster than the outer-side wheels. This contradicts our analysis. The inner wheels travel less distance and must move slower, not faster, than the outer wheels.

Therefore, the correct statement is that the wheels closer to the inside of the track move slower than the outer-side wheels.