A trolley with two wheels one metre apart is moved clockwise on the circular track around a ground with radius 50 m (described by right wheel). If the size of each wheel is of 1 foot radius and the right wheel turns 1000 times, how many times will the other wheel turn?

To solve the problem, we need to compare the number of turns made by the right wheel and the left wheel while the trolley moves around the circular track.

### Given:
1. The radius of the circular track (right wheel's path) is 50 meters.
2. The distance between the two wheels is 1 meter (this is the distance between the tracks of the right and left wheels).
3. Each wheel has a radius of 1 foot (approximately 0.3048 meters).
4. The right wheel turns 1000 times.

#### Step 1: Calculate the circumference of the right wheel's path.
The right wheel is moving along the circular path with a radius of 50 meters. The circumference \( C_{\text{right track}} \) of the circle is given by:
\[
C_{\text{right track}} = 2\pi \times 50 = 100\pi \text{ meters}
\]

#### Step 2: Calculate the total distance traveled by the right wheel.
The right wheel turns 1000 times. The circumference of the right wheel itself \( C_{\text{wheel}} \) is:
\[
C_{\text{wheel}} = 2\pi \times 0.3048 \approx 1.915 \text{ meters}
\]
The total distance \( D_{\text{right}} \) traveled by the right wheel is:
\[
D_{\text{right}} = 1000 \times 1.915 = 1915 \text{ meters}
\]

Since the distance traveled by the right wheel on the ground is equal to the circumference of the track, the trolley moves along the entire right wheel path (the outer circle). Therefore, the right wheel covers the track's circumference \( C_{\text{right track}} = 100\pi \approx 314.16 \text{ meters} \).

#### Step 3: Calculate the circumference of the left wheel's path.
The left wheel is 1 meter inside the right wheel. So, the radius of the left wheel's path is \( 50 - 1 = 49 \text{ meters} \). The circumference \( C_{\text{left track}} \) of the left wheel's path is:
\[
C_{\text{left track}} = 2\pi \times 49 = 98\pi \approx 307.88 \text{ meters}
\]

#### Step 4: Calculate the number of turns made by the left wheel.
The total distance traveled by the left wheel is \( C_{\text{left track}} \). The number of turns \( N_{\text{left}} \) made by the left wheel is:
\[
N_{\text{left}} = \frac{\text{Distance traveled by the left wheel}}{\text{Circumference of the left wheel}}
\]
The circumference of the left wheel is the same as the right wheel, \( C_{\text{wheel}} = 1.915 \text{ meters} \). Therefore, the number of turns made by the left wheel is:
\[
N_{\text{left}} = \frac{307.88}{1.915} \approx 160.71 \text{ turns}
\]

Thus, the left wheel will turn approximately **160.71 times** when the right wheel turns 1000 times.

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Do you want more details or have any questions? 

Here are some related questions to explore further:
1. How would the result change if the wheels had different radii?
2. What happens if the trolley is moved counterclockwise?
3. How does the distance between the wheels affect the number of turns?
4. What would happen if the radius of the track is doubled?
5. How would the number of turns change if both wheels were turning on a straight path?

**Tip:** When dealing with circular motion problems, always focus on comparing the circumferences and radii of paths to calculate relative distances and rotations.

Learn how to calculate the number of turns made by a wheel in a trolley moving on a circular track. The problem involves a trolley with two wheels one meter apart, moving clockwise on a 50-meter radius circular track, where the right wheel turns 1000 times. Understand key mathematical concepts such as circular motion, geometry, and ratios. Suitable for students in Grades 9-12.

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