A person is painting his house walls. He stands on a ladder with a bucket contaiing paint in one hand and a brush in other.Suddenly the bucket slips from his hand and falls down on the floor. If the bucket with the paint had a mass of 6.0 kg and was at a height of 2.0 m at the time it slipped, how much gravitational potential energy is lost together with the paint?

Explanation

To solve the problem of how much gravitational potential energy is lost when the bucket of paint falls, we can follow these steps:

### Step 1: Understand the formula for gravitational potential energy (GPE)
The gravitational potential energy (GPE) of an object at a height \( h \) is given by the formula:
\[ \text{GPE} = m \cdot g \cdot h \]
where:
- \( m \) = mass of the object (in kg)
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) = height above the ground (in meters)

### Step 2: Identify the values given in the problem
From the problem, we have:
- Mass of the bucket \( m = 6.0 \, \text{kg} \)
- Height \( h = 2.0 \, \text{m} \)
- Acceleration due to gravity \( g \approx 9.81 \, \text{m/s}^2 \)

### Step 3: Calculate the initial gravitational potential energy
Using the formula for GPE, we can calculate the initial gravitational potential energy when the bucket is at a height of 2.0 m:
\[ \text{GPE}_{\text{initial}} = m \cdot g \cdot h = 6.0 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 \cdot 2.0 \, \text{m} \]
Calculating this gives:
\[ \text{GPE}_{\text{initial}} = 6.0 \cdot 9.81 \cdot 2.0 = 117.72 \, \text{J} \]

### Step 4: Calculate the final gravitational potential energy
When the bucket falls to the ground, its height becomes 0 m. Therefore, the final gravitational potential energy is:
\[ \text{GPE}_{\text{final}} = m \cdot g \cdot 0 = 0 \, \text{J} \]

### Step 5: Calculate the loss of gravitational potential energy
The loss of gravitational potential energy is given by:
\[ \text{Loss of GPE} = \text{GPE}_{\text{final}} - \text{GPE}_{\text{initial}} = 0 - 117.72 \, \text{J} = -117.72 \, \text{J} \]
The negative sign indicates a loss of energy.

### Final Answer
The gravitational potential energy lost together with the paint is approximately **117.72 Joules**.

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