You hold a bucket attached to a rope, and in the bucket is a 500.0 g rock. You swing the bucket so the rock moves in a vertical circle 1.6 m in diameter.
What is the minimum speed the rock must have at the top of the circle if it is to always stay in contact with the bottom of the bucket?
To find the minimum speed the rock must have at the top of the circle to always stay in contact with the bottom of the bucket, we need to consider the forces acting on the rock.
At the top of the circle, the rock is moving in a vertical circle, experiencing two main forces: the tension force from the rope and its weight. The tension force provides the centripetal force required to keep the rock moving in a circle.
Let's break down the forces at the top of the circle:
1. Weight (mg): The weight of the rock always acts straight downward toward the center of the Earth, with a magnitude of mg, where m is the mass of the rock (500.0 g = 0.5 kg) and g is the acceleration due to gravity (9.8 m/s²).
2. Tension in the rope (T): The tension force acts inwards along the radius of the circular motion. At the top of the circle, the tension force should be at its minimum to prevent the rock from losing contact with the bucket.
The tension force at the top of the circle can be calculated using the following equation:
T = mg + mv²/r,
where v is the velocity of the rock at the top of the circle and r is the radius of the circle (1.6 m).
Since we want to find the minimum speed, we can assume that the tension force is just enough to provide the required centripetal force (i.e., zero net force acting towards the center) to keep the rock moving in a circle.
Setting the net force equal to zero, we have:
0 = mg + mv²/r.
Simplifying the equation, we get:
v² = rg.
Now, we can substitute the values:
v² = (1.6 m)(9.8 m/s²),
v² = 15.68 m²/s².
Finally, to find the minimum speed, we take the square root of both sides:
v = √(15.68 m²/s²),
v ≈ 3.96 m/s.
Therefore, the minimum speed the rock must have at the top of the circle in order to always stay in contact with the bottom of the bucket is approximately 3.96 m/s.