Sure, let's solve this step-by-step.
1. Understanding the Gravitational Potential Energy (PE) Formula:
The gravitational potential energy (PE) is calculated using the formula:
[tex]\[ PE = m \times g \times h \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the object (in kilograms),
- [tex]\( g \)[/tex] is the acceleration due to gravity (in [tex]\( N/kg \)[/tex]), and
- [tex]\( h \)[/tex] is the height above the ground (in meters).
2. Calculate the Gravitational Potential Energy of the Book:
- Mass of the book, [tex]\( m_{book} = 0.35 \)[/tex] kg
- Height of the shelf, [tex]\( h_{book} = 2.0 \)[/tex] m
- Acceleration due to gravity, [tex]\( g = 9.8 \)[/tex] [tex]\( N/kg \)[/tex]
Plugging these values into the formula:
[tex]\[ PE_{book} = 0.35 \text{ kg} \times 9.8 \text{ N/kg} \times 2.0 \text{ m} \][/tex]
3. Simplify the Calculation for the Book's PE:
[tex]\[ PE_{book} = 0.35 \times 9.8 \times 2.0 \][/tex]
[tex]\[ PE_{book} = 6.86 \text{ joules} \][/tex]
So, the gravitational potential energy of the book is 6.86 joules.
4. Use the Book's PE to Find the Height for the Frame:
We need the picture frame to have the same gravitational potential energy of 6.86 joules. Let's find the height at which the frame must be placed to have this energy.
- Mass of the frame, [tex]\( m_{frame} = 0.5 \)[/tex] kg
- We want [tex]\( PE_{frame} = PE_{book} = 6.86 \)[/tex] joules
- Using the formula [tex]\( PE = m \times g \times h \)[/tex], we need to solve for [tex]\( h \)[/tex]:
[tex]\[ 6.86 = 0.5 \text{ kg} \times 9.8 \text{ N/kg} \times h \][/tex]
Rearrange to solve for height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{6.86}{0.5 \times 9.8} \][/tex]
5. Simplify the Calculation for the Frame's Height:
[tex]\[ h = \frac{6.86}{4.9} \][/tex]
[tex]\[ h = 1.4 \text{ meters} \][/tex]
Therefore, the picture frame will have the same gravitational potential energy as the book when it's raised to a height of 1.4 meters above the ground.
