Gravitation Near Earth's Surface — Question 6

Question 6

A vertical acceleration can conceal the gravity change with altitude

At altitude $h=.02R$, a rocket platform accelerates radially outward at $a=.05g_0$. A passenger of mass $m$ stands on a scale. Ignore rotation and atmosphere.

1. Find the scale reading in units of $mg_0$.

2. Find the outward platform acceleration that would instead make the scale read exactly $mg_0$.

Question 6 - Solution

A vertical acceleration can conceal the gravity change with altitude

(a) Derivation and calculation

For outward positive, the passenger’s force balance is $$N-mg(h)=ma,\qquad g(h)=\frac{g_0}{(1+h/R)^2}.$$ Therefore $$\frac N{mg_0}=\frac1{1.02^2}+.05 =\boxed{1.01116878}.$$ The support force exceeds surface weight despite the lower local gravity.

(b) Result and interpretation

Demanding $N=mg_0$ gives $$a=g_0-g(h),\qquad \boxed{\frac a{g_0}=1-\frac1{1.02^2}=0.03883122}.$$ A single scale reading measures $g(h)+a$; it cannot separately determine these two contributions.

Physical check

Setting $a=-g(h)$ gives $N=0$, consistent with free fall.