Question

gravitational potential energy
a 3 kg book is lifted to a shelf 2 meters high. what is its gravitational potential energy?
a 10 kg box is raised 1.5 meters off the ground. find its gravitational potential energy.
a 2.5 kg rock is on a hill 8 meters high. how much gravitational potential energy does it have?
a 15 kg backpack is lifted 0.5 meters onto a table. calculate its gravitational potential energy.
elastic potential energy
a spring with a spring constant of 100 n/m is stretched 0.2 m. find the elastic potential energy.
a rubber band with k=50n/m is stretched 0.1 m. how much energy is stored?
a trampoline spring has k=250n/m and stretches 0.05 m. find the elastic potential energy.

Explanation:

Step1: Gravitational Potential Energy Formula

The formula for gravitational potential energy is \(U = mgh\), where \(m\) is mass (\(kg\)), \(g = 9.8\ m/s^{2}\) (acceleration due to gravity), and \(h\) is height (\(m\)).

Step2: Calculate for the book

For the \(3\ kg\) book (\(m = 3\ kg\), \(h = 2\ m\)):
\(U_1=3\times9.8\times2= 58.8\ J\)

Step3: Calculate for the box

For the \(10\ kg\) box (\(m = 10\ kg\), \(h = 1.5\ m\)):
\(U_2 = 10\times9.8\times1.5=147\ J\)

Step4: Calculate for the rock

For the \(2.5\ kg\) rock (\(m = 2.5\ kg\), \(h = 8\ m\)):
\(U_3=2.5\times9.8\times8 = 196\ J\)

Step5: Calculate for the backpack

For the \(15\ kg\) backpack (\(m = 15\ kg\), \(h = 0.5\ m\)):
\(U_4=15\times9.8\times0.5 = 73.5\ J\)

Step6: Elastic Potential Energy Formula

The formula for elastic potential energy is \(U_e=\frac{1}{2}kx^{2}\), where \(k\) is the spring constant (\(N/m\)) and \(x\) is the displacement (\(m\)).

Step7: Calculate for the spring

For the spring (\(k = 100\ N/m\), \(x = 0.2\ m\)):
\(U_{e1}=\frac{1}{2}\times100\times(0.2)^{2}= 2\ J\)

Step8: Calculate for the rubber - band

For the rubber - band (\(k = 50\ N/m\), \(x = 0.1\ m\)):
\(U_{e2}=\frac{1}{2}\times50\times(0.1)^{2}=0.25\ J\)

Step9: Calculate for the trampoline spring

For the trampoline spring (\(k = 250\ N/m\), \(x = 0.05\ m\)):
\(U_{e3}=\frac{1}{2}\times250\times(0.05)^{2}=0.3125\ J\)

Answer:

The gravitational potential energies are: \(58.8\ J\) (book), \(147\ J\) (box), \(196\ J\) (rock), \(73.5\ J\) (backpack). The elastic potential energies are: \(2\ J\) (spring), \(0.25\ J\) (rubber - band), \(0.3125\ J\) (trampoline spring).