Question

When an 89.2-kg adult uses a spiral staircase to climb to the second floor of his house, his gravitational potential energy increases by 2.29 x 103 J. By how much does the potential energy of a 17.5-kg child increase when the child climbs a normal staircase to the second floor?

Answer 1

Answer 2

To calculate the increase in potential energy for the 17.5-kg child climbing a normal staircase to the second floor, we can use the same principle of gravitational potential energy.

The formula for gravitational potential energy is:

PE = m * g * h

where: PE = Gravitational potential energy m = Mass of the object (in kg) g = Acceleration due to gravity (approximately 9.81 m/s^2 on Earth) h = Height (vertical distance) gained (in meters)

We already know the gravitational potential energy increase for the adult is 2.29 x 10^3 J. Let's assume the height gained by both the adult and the child is the same (the height of climbing to the second floor).

Let h be the height gained by climbing to the second floor in meters.

For the adult: PE_adult = 89.2 kg * 9.81 m/s^2 * h

For the child: PE_child = 17.5 kg * 9.81 m/s^2 * h

Since the potential energy increase for the child is not given, we'll denote it as ΔPE_child. Therefore, we have:

ΔPE_child = PE_child - Initial_PE_child

where Initial_PE_child is the initial potential energy of the child on the ground level, which is zero (we take the ground level as the reference point for potential energy).

Now, we know the gravitational potential energy increase for the adult is 2.29 x 10^3 J, so:

2.29 x 10^3 J = 89.2 kg * 9.81 m/s^2 * h

From this equation, we can solve for h:

h = (2.29 x 10^3 J) / (89.2 kg * 9.81 m/s^2)

h ≈ 2.58 meters

Now, we can calculate the potential energy increase for the child:

ΔPE_child = 17.5 kg * 9.81 m/s^2 * h

ΔPE_child = 17.5 kg * 9.81 m/s^2 * 2.58 m

ΔPE_child ≈ 450.62 J

So, the potential energy of the 17.5-kg child increases by approximately 450.62 J when the child climbs a normal staircase to the second floor.

Answer 3

To find the change in potential energy for the 17.5-kg child climbing a normal staircase to the second floor, we can use the same concept of gravitational potential energy as we did for the adult.

The formula for gravitational potential energy is:

Potential Energy (PE) = m * g * h

where: m = mass of the object (in kilograms) g = acceleration due to gravity (approximately 9.81 m/s² on Earth) h = height or vertical displacement (in meters)

For the adult: Mass (m1) = 89.2 kg Change in Potential Energy (ΔPE1) = 2.29 x 10^3 J

For the child: Mass (m2) = 17.5 kg Change in Potential Energy (ΔPE2) = ?

Since the height or vertical displacement (h) is the same for both the adult and the child (climbing to the second floor), we can set up the following equation:

m1 * g * h = ΔPE1 m2 * g * h = ΔPE2

We know the values of m1, ΔPE1, and m2, and we can use the value of g (approximately 9.81 m/s²) to find ΔPE2:

ΔPE2 = (m2 / m1) * ΔPE1

ΔPE2 = (17.5 kg / 89.2 kg) * 2.29 x 10^3 J

ΔPE2 ≈ 0.45 x 10^3 J

ΔPE2 ≈ 450 J

So, the potential energy of the 17.5-kg child increases by approximately 450 Joules when the child climbs a normal staircase to the second floor.