To solve this problem, we need to analyze the motion of the umbrella and how it affects the trajectory of the raindrops falling from it. The umbrella is being whirled around at a uniform rate, and we can use some basic principles of circular motion to determine the radius of the circle traced by the drops on the floor.
Understanding the Setup
We have an umbrella with a diameter of 1 meter, which means its radius is:
- Radius of the umbrella (r) = Diameter / 2 = 1 m / 2 = 0.5 m
The height of the rim of the umbrella above the floor is 4.9 meters. As the umbrella is whirled around, the drops of rain will fall vertically downwards from the rim of the umbrella. The key here is to find out how far from the center of the umbrella's circular path the drops will land on the floor.
Calculating the Radius of the Locus of Drops
When the umbrella is spun, it completes half a revolution every second. This means that in one second, the umbrella moves through an angle of 180 degrees. The drops will fall straight down while the umbrella moves in a circular path.
Finding the Radius of the Circle on the Floor
As the umbrella is whirled, the drops will land on the floor in a circular pattern. The radius of this circle (R) can be determined by considering the horizontal distance from the center of the umbrella to the point where the drops hit the ground.
Since the umbrella is held at a height of 4.9 meters and is spinning, we can visualize the situation as follows:
- The radius of the umbrella (0.5 m) is the distance from the center of the umbrella to the edge.
- The drops fall vertically downwards from the edge of the umbrella.
When the umbrella is at its maximum horizontal displacement during the spin, the drops will land at a distance equal to the radius of the umbrella plus the radius of the circular path traced by the center of the umbrella. The center of the umbrella moves in a circle of radius equal to the height of the rim above the floor (4.9 m).
Final Calculation
The total radius (R) of the circle on the floor where the drops land can be calculated as follows:
- R = Height of the rim + Radius of the umbrella
- R = 4.9 m + 0.5 m = 5.4 m
Thus, the radius of the locus of drops on the floor is 5.4 meters.
Visualizing the Motion
To further clarify, imagine standing in the center of a large circle with the umbrella at the edge. As you spin the umbrella, the drops fall straight down, but because the umbrella is moving, the drops land in a circular pattern on the floor. The height of the umbrella and the radius of the umbrella combine to create this larger circle where the drops fall.
In summary, by understanding the relationship between the height of the umbrella and its radius, we can determine the radius of the circle on the floor where the raindrops will land. In this case, that radius is 5.4 meters.
