To solve the problem, we need to find the height at which a picture frame has the same gravitational potential energy as a book kept on a shelf. We'll use the formula for gravitational potential energy, [tex]\( PE = m \times g \times h \)[/tex], where [tex]\( PE \)[/tex] is the potential energy, [tex]\( m \)[/tex] is the mass, [tex]\( g \)[/tex] is the gravitational acceleration, and [tex]\( h \)[/tex] is the height.
Let's go through the steps:
1. Calculate the gravitational potential energy of the book:
- Mass of the book, [tex]\( m_{book} = 0.35 \)[/tex] kg
- Height of the book above ground, [tex]\( h_{book} = 2.0 \)[/tex] meters
- Gravitational acceleration, [tex]\( g = 9.8 \)[/tex] N/kg
Using the formula:
[tex]\[
PE_{book} = m_{book} \times g \times h_{book}
\][/tex]
Substituting the values:
[tex]\[
PE_{book} = 0.35 \times 9.8 \times 2.0
\][/tex]
Calculate the product:
[tex]\[
PE_{book} = 6.86 \text{ Joules}
\][/tex]
2. Determine the height at which the picture frame must be raised to have the same potential energy:
- Mass of the picture frame, [tex]\( m_{frame} = 0.5 \)[/tex] kg
- We need to find the height [tex]\( h_{frame} \)[/tex] so that [tex]\( PE_{frame} = PE_{book} \)[/tex]
Using the formula for potential energy again, but rearranging to solve for height:
[tex]\[
PE_{frame} = m_{frame} \times g \times h_{frame}
\][/tex]
Since we want [tex]\( PE_{frame} = PE_{book} \)[/tex], we can set:
[tex]\[
6.86 = 0.5 \times 9.8 \times h_{frame}
\][/tex]
Solving for [tex]\( h_{frame} \)[/tex]:
[tex]\[
h_{frame} = \frac{6.86}{0.5 \times 9.8}
\][/tex]
Calculate the denominator:
[tex]\[
0.5 \times 9.8 = 4.9
\][/tex]
Now, divide:
[tex]\[
h_{frame} = \frac{6.86}{4.9} \approx 1.4 \text{ meters}
\][/tex]
So, the picture frame must be raised to a height of 1.4 meters to have the same gravitational potential energy as the book.
