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23P3Q07

7a)

Electric field strength at a point is the force per unit positive charge acting on a small test charge

placed at that point.

7bi)

COMMENT: Equally space lines, arrows pointing inwards, ending perpendicularly on the surface of the sphere.

7bii)

\displaystyle \begin{aligned}  & (E=k\frac{Q}{{{r}^{2}}}) \\ & 130=(8.99\times {{10}^{9}})\frac{Q}{{{(0.15+0.40)}^{2}}} \\ & =4.37\times {{10}^{-9}}\text{ C} \end{aligned}

23P3Q06

6ai)

1

COMMENT: The mean KE is directly proportional to the kelvin temperature.

6aii)

0.25

COMMENT: Think \displaystyle \frac{PV}{T}=nR

6aiii)

Since they have the same mean KE,

\displaystyle \begin{aligned}  & \frac{1}{2}(m){{c}_{A}}^{2}=\frac{1}{2}(2m){{c}_{B}}^{2} \\ & \frac{{{c}_{A}}}{{{c}_{B}}}=\sqrt{2} \end{aligned}

6b)

R.m.s. speed of the molecules in container A, \displaystyle {{c}_{A}}=\sqrt{2}(940)=1329\text{ m }{{\text{s}}^{-1}}

Mass of a molecule in container A, \displaystyle m=\frac{{{M}_{molar}}}{{{N}_{A}}}=\frac{4.0\times {{10}^{-3}}}{6.02\times {{10}^{23}}}=6.645\times {{10}^{-27}}\text{ kg}

\displaystyle

\displaystyle \begin{aligned}  \left\langle KE \right\rangle &=\frac{3}{2}kT \\ \frac{1}{2}m{{c}_{A}}^{2}&=\frac{3}{2}kT \\ (6.645\times {{10}^{-27}}){{(1329)}^{2}}&=3(1.38\times {{10}^{-23}})T \\ T&=283\text{ K} \end{aligned}

23P3Q05

5a)

The gravitational potential at a point is the work done per unit mass

by an external force to bring a (small) mass from infinity to that point (without any change in KE of the

mass).

5b)

By principle of conservation of energy, the total KEi + GPEi when the projectile is on the surface of the planet, should be the same as the total KEf + GPEf when it is at the point at infinity.

\displaystyle \begin{aligned}  & K{{E}_{i}}+GP{{E}_{i}}=K{{E}_{f}}+GP{{E}_{f}} \\ & \frac{1}{2}m{{v}^{2}}+(-\frac{GMm}{R})=0+0 \\ & v=\sqrt{\frac{2GM}{R}} \end{aligned}

5ci)

COMMENT: GPE + KE should be equal to zero, since the projectile just manages to escape. That’s how we deduced that GPE on the surface must be \displaystyle -8.0\times {{10}^{6}}\text{ J}.

COMMENT: Remember that gravitational potential is inversely proportional to distance.

COMMENT: Since GPE + KE is constant, the GPE and KE graphs are mirror images of each other.