2023 GCE A Level 9814 H3 P1 SS [HCI]

Hwa Chong Institution (College) 2023 H3 A level Suggested Solutions 1(a)(i) The restoring force (tangential component of weight) is the net force. By Newton’s 2nd law, sinmg ma−= sin r = l where r is the amplitude of the oscillation. For small angles, sin , so the amplitude r becomes l . The motion of pendulum is approximated to be a simple harmonic motion, hence, 22ar   =− =− l Rewriting Newton’s 2nd law, ( ) 2 2 2 2 mg m g T g       − = − = == l l l 1(a)(ii) 11 2 1 9.81 0.910 Hz (3 s.f.)2 0.300 gf T f   == == l 1(b)(i) 2 2 2 2 0 0 0 1 0 LCVV di qL dt C d q qL dt C dq qdt LC += += += += The differential equation for q is analogous to the simple harmonic motion. θ mg Tension mgsinθ m l r

Hwa Chong Institution (College) ( ) 22 1 2 1 2 fLC f LC   == = 1(b)(ii) 36 11 2.81 Hz 2 2 82 10 (39000 10 ) 2.81 3.090.910 3 LC LC pendulum LC pendulum f LC f f ff   −− = = =  ==  1(c) The natural frequency of LC circuit must be 3 times smaller. So, the effective capacitance of all the capacitors should be 9 times greater. To achieve this, 9 capacitors should be connected in parallel. However, this does not exactly match, 0.938 Hz and 0.910 HzLC pendulumff == . To exactly match the frequencies , connect 10 capacitors in series this gives 0.890 HzLCf = and increase the length of the pendulum to 31.4 cm, which gives 0.890 Hzpendulumf = .

Hwa Chong Institution (College) 2(a) The magnitude of the electric dipole moment is the product of charge on an atom and distance between the centres of positive and negative charges. 2(b) 30 20 12 20 19 1.50 3.34 10 6.67 10 C(0.784)(95.8 10 ) 6.67 10 C 1.60 10 C 0.417 pq d e qe − − − − − = = =     = 2(c) By applying cosine rule, 22(1.50D) (1.50D) 2(1.50D)(1.50D)cos104 1.85 net net p pD = + −  = 2(d)(i) The centre of mass should be closer to the oxygen atom as it is heavier. 2(d)(ii) electric sin sin sinF r qEr pE   = = = 2(d)(iii) Initially, the dipole is stationary. It has zero rotational kinetic energy. Its potential energy is cos(63 )pE  . 104° 1.50D 1.50D Anti-clockwise Felectric Felectric

Hwa Chong Institution (College) The dipole will have maximum kinetic energy when it has minimum potential energy (the dipole vector is parallel with the electric field, hence the angle is zero). The potential energy at this moment is cos(0 )pE  . The maximum rotational kinetic energy would be the change in potential energy between its initial state and when the dipole moment vector is parallel with the electric field. 34 26 cos(0 ) cos(63 ) 1.85 3.34 10 9500 (1 cos(63 )) 3.21 10 J pE pE − − =  −  =     −  =

Hwa Chong Institution (College) 3(a) 3(b) The net force in vertical direction: cos sin 0mg− − =NF (1) The net force in horizontal direction: 2 maxcos sin mv r+=FN (2) The frictional force is N . Substituting this into (1):