Two 0.61 kg basketballs, each with a radius of 13 cm, are just touching. How much energy is required to change the separation between the centers of the basketballs to the following (a)1 meter
(b) 10 meters
the gravitational constant G =6.67•10⁻¹¹ N•m²/kg²,
PE₁=G•m₁•m₂/R₁
PE₂=G•m₁•m₂/R₂
E=ΔPE= PE₁-PE₂=
=G•m₁•m₂/R₁-G•m₁•m₂/R₂=
=G•m₁•m₂(1/ R₁- 1/R₂)= …
R₁=0.26 m
R₂= 1 m, 10 m
To calculate the energy required to change the separation between the centers of the basketballs, we need to first calculate the gravitational potential energy between them.
The gravitational potential energy between two objects can be calculated using the formula:
E = -G * ((m1 * m2) / r)
Where:
E = Gravitational potential energy
G = Universal gravitational constant (approximately 6.67430 × 10^-11 N m² / kg²)
m1 and m2 = Masses of the basketballs
r = Separation between the centers of the basketballs
Given:
Mass of each basketball (m1 and m2) = 0.61 kg
Radius of the basketballs = 13 cm = 0.13 m
To calculate the separation (r) between the centers of the basketballs:
Since the basketballs are just touching, the separation between their centers is equal to twice the radius.
a) For a separation of 1 meter:
r = 2 * radius = 2 * 0.13 m = 0.26 m
Using the given mass and separation, we can calculate the energy required:
E1 = -G * ((m1 * m2) / r)
= - (6.67430 × 10^-11 N m² / kg²) * ((0.61 kg * 0.61 kg) / 0.26 m)
= -3.072 × 10^-10 J
Therefore, the energy required to change the separation between the centers of the basketballs to 1 meter is approximately -3.072 × 10^-10 Joules.
b) For a separation of 10 meters:
r = 2 * radius = 2 * 0.13 m = 0.26 m
Using the given mass and separation, we can calculate the energy required:
E2 = -G * ((m1 * m2) / r)
= - (6.67430 × 10^-11 N m² / kg²) * ((0.61 kg * 0.61 kg) / 10 m)
= -2.567 × 10^-12 J
Therefore, the energy required to change the separation between the centers of the basketballs to 10 meters is approximately -2.567 × 10^-12 Joules.