Answer:
The tension in the rope (468.19 N) is less than the maximum force the rope can withstand (525 N), the rope will be able to support the bucket at the bottom of the circle without breaking.
Step-by-step explanation:
To calculate the tension in the rope at the bottom of the circle, we need to consider the forces acting on the bucket. At the bottom of the circle, two main forces are acting on the bucket:
Tension force (T) in the rope, directed upward.
Gravitational force (weight) acting downward.
Since the bucket is moving in a circular path at a constant speed, there is also a centripetal force acting inward toward the center of the circle. In this case, the centripetal force is provided by the tension in the rope.
The tension in the rope can be calculated using the following formula:
Tension (T) = Centripetal Force
The centripetal force can be calculated using the formula:
Centripetal Force = (mass of the bucket) × (centripetal acceleration)
The centripetal acceleration can be calculated using the formula:
Centripetal acceleration = (angular velocity)^2 × (radius of the circle)
Given data:
Mass of the bucket (m) = 4.54 kg
Frequency of rotation (f) = 2.0 Hz
Length of the rope (radius of the circle, r) = 65 cm = 0.65 m
Maximum force the rope can withstand = 525 N
Step 1: Calculate the centripetal acceleration.
First, we need to find the angular velocity (ω) using the frequency:
Angular velocity (ω) = 2π × (frequency)
ω = 2π × 2.0 Hz ≈ 12.57 rad/s
Now, calculate the centripetal acceleration:
Centripetal acceleration = ω^2 × r
Centripetal acceleration = (12.57 rad/s)^2 × 0.65 m ≈ 103.20 m/s²
Step 2: Calculate the tension in the rope.
Now, we can calculate the tension (T) in the rope using the centripetal force formula:
Tension (T) = Mass × Centripetal acceleration
T = 4.54 kg × 103.20 m/s² ≈ 468.19 N
Since the tension in the rope (468.19 N) is less than the maximum force the rope can withstand (525 N), the rope will be able to support the bucket at the bottom of the circle without breaking.