A 14.0 kg bucket is lowered vertically by rope in which there is 163 N of tension at a given instant. What is the acceleration of the bucket? Is it up or down?
Understand the Problem
The question is asking us to determine the acceleration of a 14.0 kg bucket that is being lowered by a rope with a tension of 163 N. We need to use Newton's second law and consider the forces acting on the bucket to find its acceleration and direction.
Answer
$1.83 \, \text{m/s}^2$ downward.
Answer for screen readers
The acceleration of the bucket is $1.83 , \text{m/s}^2$ downward.
Steps to Solve
- Identify the forces acting on the bucket
The two primary forces acting on the bucket are the weight (force of gravity) and the tension in the rope.
- Weight ($F_g$) can be calculated as: $$ F_g = m \cdot g $$ Where:
- $m = 14.0 , \text{kg}$ (mass of the bucket)
- $g = 9.81 , \text{m/s}^2$ (acceleration due to gravity)
- Calculate the weight of the bucket
Now, let's find the gravitational force (weight) acting on the bucket: $$ F_g = 14.0 , \text{kg} \cdot 9.81 , \text{m/s}^2 = 137.34 , \text{N} $$
- Set up Newton's second law
According to Newton's second law: $$ F_{net} = m \cdot a $$ Where:
- $F_{net}$ is the net force acting on the bucket
- $a$ is the acceleration
- Determine the net force acting on the bucket
The net force can be determined by the difference between the tension ($T$) and the weight ($F_g$): $$ F_{net} = T - F_g $$ Substituting the values: $$ F_{net} = 163 , \text{N} - 137.34 , \text{N} = 25.66 , \text{N} $$
- Calculate the acceleration of the bucket
Now we can find the acceleration using our net force: $$ a = \frac{F_{net}}{m} $$ Substituting the known values: $$ a = \frac{25.66 , \text{N}}{14.0 , \text{kg}} = 1.83 , \text{m/s}^2 $$
The acceleration of the bucket is $1.83 , \text{m/s}^2$ downward.
More Information
The acceleration calculated indicates that the bucket is accelerating downward, which makes sense since it is being lowered. This value represents how quickly the bucket's speed increases as it moves downwards.
Tips
- A common mistake is forgetting to subtract the weight from the tension, leading to an incorrect net force calculation. Always remember to consider the direction of the forces.
- Another mistake is forgetting to convert units or miscalculating the values for $g$. It's important to ensure the standard value of $g$ is used, which is approximately $9.81 , \text{m/s}^2$.
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