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 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 vertical circle, the gravitational force acting on the rock is acting as the centripetal force keeping it in circular motion. To stay in contact with the bottom of the bucket, the centrifugal force acting on the rock must be less than or equal to the gravitational force acting on it.
The centripetal force can be calculated using the formula:
F_centripetal = m * v^2 / r,
where F_centripetal is the centripetal force, m is the mass of the rock (0.5 kg), v is the velocity of the rock at the top of the circle, and r is the radius of the circle (half the diameter, so 0.8 m).
The gravitational force acting on the rock can be calculated using the formula:
F_gravity = m * g,
where F_gravity is the gravitational force and g is the acceleration due to gravity (approximately 9.8 m/s^2).
To determine the minimum speed at the top of the circle, we will set the centrifugal force equal to the gravitational force:
m * v^2 / r = m * g.
Simplifying this equation gives:
v^2 = r * g.
Plugging in the values, we get:
v^2 = 0.8 m * 9.8 m/s^2.
Solving for v, we find:
v = sqrt(0.8 * 9.8) m/s.
Calculating this, we get:
v ≈ 3.14 m/s.
Therefore, the minimum speed the rock must have at the top of the circle to always stay in contact with the bottom of the bucket is approximately 3.14 m/s.
To determine 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, we need to analyze the forces acting on the rock at the topmost point.
When the rock is at the top of the circle, two forces are acting on it: the tension force provided by the rope and the force of gravity. The tension force must be sufficient to provide the necessary centripetal force to keep the rock moving in a circular path.
At the very top of the circle, the tension force is directed towards the center of the circle, while the force of gravity is directed downwards, away from the center. Therefore, the tension force must be equal to or greater than the force of gravity at this point.
First, let's calculate the force of gravity acting on the rock:
Force of gravity = mass × acceleration due to gravity
Given:
Mass of the rock (m) = 500.0 g = 0.5 kg
Acceleration due to gravity (g) = 9.8 m/s²
Force of gravity = 0.5 kg × 9.8 m/s² = 4.9 N
Since the rock must have a speed such that the tension force is equal to or greater than the force of gravity, we can set up the following equation:
Tension force ≥ Force of gravity
Let's denote the minimum required tension force as T. At the top of the circle, the tension force and the force of gravity can be related as follows:
T - Force of gravity = centripetal force
The centripetal force can be determined using the following equation:
Centripetal force = (mass × velocity²) / radius
We know the mass, radius, and force of gravity. Therefore, we can rearrange the equation to solve for velocity squared:
T - Force of gravity = (mass × velocity²) / radius
T - 4.9 N = (0.5 kg × velocity²) / 0.8 m (assuming the given diameter is the actual distance traveled; so the radius is 0.8m)
Now we need to calculate the tension force T. At the top of the circle, the tension force can be determined using the following equation:
T = mass × (gravity + centripetal acceleration)
The centripetal acceleration is given by:
Centripetal acceleration = velocity² / radius
T = 0.5 kg × (9.8 m/s² + (velocity² / 0.8 m))
Now we can substitute this value of T into the previous equation:
0.5 kg × (9.8 m/s² + (velocity² / 0.8 m)) - 4.9 N = (0.5 kg × velocity²) / 0.8 m
Now we can solve this equation to find the minimum velocity required for the rock to stay in contact with the bucket at the top of the circle.