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 stay in contact with the bottom of the bucket, we can use the concept of centripetal force. At the top of the circle, the net force acting on the rock must be equal to the centripetal force required to keep it moving in a circle.
1. First, we need to calculate the force of gravity acting on the rock. The weight of the rock can be calculated using the formula: weight = mass * acceleration due to gravity.
weight = 0.500 kg * 9.8 m/s^2
weight = 4.9 N
2. At the top of the circle, the net force acting on the rock is the sum of the gravitational force (downward) and the tension force (upward) in the rope. The tension force provides the centripetal force required to keep the rock moving in a circle.
net force = tension force - weight
3. To stay in contact with the bottom of the bucket, the net force acting on the rock at the top of the circle must be equal to the centripetal force required.
net force = centripetal force
4. The centripetal force is given by the equation: centripetal force = mass * (velocity^2 / radius)
5. Since the rock is moving in a vertical circle, the radius of the circle is half the diameter.
radius = 1.6 m / 2
radius = 0.8 m
6. Now we can set up the equation using the given values.
tension force - weight = mass * (velocity^2 / radius)
7. Rearrange the equation to solve for velocity.
velocity^2 = (tension force - weight) * (radius / mass)
velocity^2 = (tension force - 4.9 N) * (0.8 m / 0.500 kg)
8. To find the minimum speed, we need to consider the scenario when the tension force is at its minimum. This happens when the rock reaches the top of the circle and the tension force becomes zero.
velocity^2 = (-4.9 N) * (0.8 m / 0.500 kg)
9. Calculate the minimum velocity.
velocity^2 = -7.84 m^2/s^2
velocity = sqrt(-7.84 m^2/s^2)
The minimum speed required for the rock to always stay in contact with the bottom of the bucket at the top of the circle is not physically possible because the result of taking the square root of a negative number is imaginary. Thus, it is not possible to keep the rock in contact with the bottom of the bucket at the top of the circle.
To determine 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 can use the concept of centripetal force.
At the topmost point of the circle, the rock is momentarily at rest before changing direction. At this point, only the gravitational force acts on the rock, which provides the necessary centripetal force to keep it in circular motion. The tension in the rope provides this centripetal force.
The centripetal force required for the rock to stay in circular motion at the top can be calculated using the equation:
F_c = m * (v^2 / r)
Where:
F_c is the centripetal force
m is the mass of the rock (500.0 g or 0.5 kg)
v is the velocity of the rock
r is the radius of the circular path (half the diameter, 0.8 m)
At the topmost point, the gravitational force acting on the rock is m * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Therefore, to ensure the rock remains in contact with the bottom of the bucket, the following condition must be met:
F_c = m * (v^2 / r) = m * g
Rearranging the equation, we can solve for the minimum velocity v:
v^2 = r * g
v = √(r * g)
Plugging in the values, we can calculate the minimum velocity:
v = √(0.8 m * 9.8 m/s^2)
v ≈ 2.8 m/s
Therefore, the minimum speed the rock must have at the top of the circle is approximately 2.8 m/s to always stay in contact with the bottom of the bucket.