如何计算万有引力？

我们在日常生活中目睹了几个重力与力一起工作的例子。我们知道，宇宙中有持续的运动和力量，以推和拉的形式存在。在无数的运动中，有四大基本力量负责各种各样的现象。引力、强力、弱力和电磁力是四种力。

重力是将物品吸引到地面的力量。力量有能力发挥作用，因此重力从根本上起到了作用。当您对某个项目施加力时，力会为您完成工作。例如，当您抛球时，施加在球上的力会使球移动一段距离，因此这项工作就完成了。功与所施加的力和由此而行进的距离成正比；例如，如果你用较小的力量扔球，则球所经过的距离会更小，与所施加的力成正比；同样地，如果你用很大的力量扔球，所覆盖的距离会很长。当一个粒子下落时，它被迫指向重力方向。

地心引力

万有引力是一种吸引宇宙中任何两个物体的力，无论它们是否具有相同的质量。此外，牛顿的万有引力定律指出，包括你在内的每一件事物都会拉动宇宙中的所有其他物体。万有引力的单位是牛顿，记为 N。

公式

F_g = $\frac{Gm_1m_2}{r^2}$

where,

• F_g is the gravitational force between the objects
• G refers to the gravitational constant, given by G = 6.67 ×10⁻¹¹N⋅ m²/kg²
• m1 and m2 depict the masses of the given objects
• r is the distance between the given objects in meters

推导

We are well aware of the fact that force is directly proportional to the product of the masses of the bodies. Hence,

F ∝  m₁m₂

Also, F is indirectly proportional to the square of the distance between the two bodies, we have:

 F ∝  $\frac{1}{d^2}$

From equations (1) and (2), we have:

 F ∝  $\frac{m_1m_2}{d^2}$

Removing the proportionality sign, we obtain:

F_g = $\frac{Gm_1m_2}{r^2}$

Hence proved.

示例问题

问题 1. 求两只大象之间的万有引力，一只质量为 1000 kg，另一只质量为 800 kg，如果它们之间的距离为 5 m。

解决方案：

Given: m₁ = 1000 kg, m₂ = 800 kg, r = 5 m

The formula for gravitational force is given as: F_g = $\frac{Gm_1m_2}{r^2}$

Here, G =  6.67 ×10⁻¹¹N⋅ m²/kg²

Substituting the values in the formula, we have:

F_g = $\frac{6.67 ×10^{−11}N⋅ m^2/kg^2)(1000 kg)(800 kg)}{5^2}$

F_g = 2.1 × 10^-6 N

问题 2. 求质量为 50 kg 的人与质量为 1500 kg 的公共汽车之间的万有引力，如果它们之间的距离为 10 m。

解决方案：

Given: m₁ = 50 kg, m₂ = 1500 kg, r = 10 m

The formula for gravitational force is given as: F_g = $\frac{Gm_1m_2}{r^2}$

Here, G =  6.67 ×10⁻¹¹N⋅ m²/kg²

Substituting the values in the formula, we have:

F_g = $\frac{6.67 ×10^{−11}N⋅ m^2/kg^2)(50 kg)(1500 kg)}{10^2}$

F_g = 5.0025 × 10^-8 N

问题 3. 地球对月球的吸引力是否与月球对地球的吸引力相等？解释。

解决方案：

The universal law of gravitation states that two objects attract each other with equal force, but in opposite directions. Hence the earth attracts the moon with an equal amount of force with which the latter attracts the former.

问题 4. 假设两个物体在一定距离处的万有引力为 4N。如果它们之间的距离加倍，求吸引力。

解决方案：

Newton’s law of gravitation states that the gravitational force between two point like objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

F_g = $\frac{Gm_1m_2}{r^2}$

This equation shows that ,for given masses, if r is replaced by 2r, the force becomes 1/4th original force. Therefore ,force of attraction will become 4/4 =1 N.

问题 5. 地球的质量是 6 × 10 ²⁴ kg。地球与太阳之间的距离为 1.5 × 10 ¹¹ m。如果两者之间的万有引力为 3.5 × 10 ²² N，那么太阳的质量是多少？

解决方案：

Given: m_e = 6 × 10²⁴ kg, r = 1.5 × 10¹¹ m and F = 3.5 × 10²² N

The formula for gravitational force is given as: Fg = $\frac{Gm_1m_2}{r^2}$.

⇒ 3.5 × 10²² N = $\frac{6.67×10^{-11}×6×10^{24}×m_{sun}}{(1.5×10^{11})^2}$

⇒ Mass of sun = $\frac{3.5×10^{22}×2.25×10^{22}}{40.02×10^{13}}$

= 1.967 × 10³⁰ kg

问题 6. 如果月球吸引地球，那为什么地球不向月球移动？

解决方案：

The earth and moon experience equal gravitational forces from each other. However the mass of earth is much larger than that of the moon. Thus it accelerates at a much lesser rate than that of the acceleration of moon towards the earth. This is why the earth moves towards the earth.