Question

If both the mass and radius of the earth decrease by $1$% the value of

• A. acceleration due to gravity would decrease by nearly $1$%
• B. acceleration due to gravity would increase by $1$%
• C. escape velocity from the earth's surface would decrease by $1$%
• D. the gravitational potential energy of a body on earth's surface will remain unchanged

Answer 1

Answer: (B), (D)

The correct options are
B acceleration due to gravity would increase by $1$%
D the gravitational potential energy of a body on earth's surface will remain unchanged

The acceleration due to gravity is: $g=\frac{GM}{R^2}$

The new value of g would be: $g^{\prime }=\frac{G\left(0.99M\right)}{(0.99R{)}^2}\simeq 1.01\frac{GM}{R^2}=1.01g$

Thus,g would increase by about 1%.The new escape velocity would be:

${V_e}^{\prime }=\sqrt{\frac{2\times 0.99M\times G}{0.99R}}=\sqrt{\frac{2MG}{R}}=V_e$

Thus,the escape velocity will remain unchanged.The potential energy of a body of mass m on earth's surface would be

$-\frac{GM\left(0.99M\right)}{\left(0.99R\right)}=-\frac{GmM}{R}$

Thus,the potential energy will also remain unchanged.Hence,the correct choices are (b) and (d).