• Physics 9702 Doubts | Help Page 182

Question 892: [Gravitation]

(a) Define gravitational potential.

(b) Explain why values of gravitational potential near to an isolated mass are all negative.

(c) Earth may be assumed to be an isolated sphere of radius 6.4 × 10³ km with its mass of 6.0 × 10²⁴ kg concentrated at its centre. An object is projected vertically from the surface of the Earth so that it reaches an altitude of 1.3 × 10⁴ km.

Calculate, for this object,

(i) change in gravitational potential,

(ii) speed of projection from the Earth’s surface, assuming air resistance is negligible.

(d) Suggest why the equation

v² = u² + 2as

is not appropriate for calculation in (c)(ii).

Reference: Past Exam Paper – June 2003 Paper 4 Q1

Solution 892:

(a) Gravitational potential (at a point) is defined as the work done in bringing/moving unit mass from infinity to the point.

(b) The potential at infinity is defined as being zero. The forces are always attractive, so work got out in moving to point (work is done on the mass when moving it to infinity).

(c)

(i)

Gravitational potential, φ = - GM / R = - GM × (1/R)

{The distances should be converted in metre. Initial position is at the surface of the Earth, which is a distance of 6.4×10⁶m from the centre of the Earth. The final distance is 1.3×10⁷m (altitude = distance above surface) + 6.4×10⁶m (distance of surface of the centre) = 1.94×10⁷m}

Change in potential = (6.67×10^-11) (6.0×10²⁴) × ({6.4×10⁶}^-1 – {1.94×10⁷}^-1)

Change in potential = 4.19 × 10⁷ J kg^-1 (ignore sign)

(ii)

The kinetic energy is converted to gravitational potential energy as the height of the object from the surface of the Earth increases.

½ mv² = mΔφ

v² = 2 × 4.19×10⁷ = 8.38 × 10⁷

Speed v = 9150 m s^-1

(d) The acceleration is not constant.

Question 893: [Kinematics > Graph]

Graph shows how the velocity v of a firework rocket changes with time t.

At which point on the graph does the rocket have the greatest acceleration?

Reference: Past Exam Paper – November 2013 Paper 13 Q7

Solution 893:

Answer: B.

The gradient of a velocity-time graph gives the acceleration.

Thus, the acceleration is greatest where the value of the gradient is greatest. Visually, this is be identified by the steepness of the tangent at a point – the steeper the tangent, the greater is the gradient.

The tangent is steepest at point B, compared with the other points.

Question 894: [Dynamics]

When a golfer hits a ball his club is in contract with the ball for about 0.0005 s and the ball leaves the club with a speed of 70 m/s. The mass of the ball is 46 g. Determine the mean force on the ball?

Reference: ???

Solution 894:

From Newton’s 2^nd law, Force F is equal / proportional to rate of change of momentum. (F = Δp / t)

Change in momentum is given by p = mΔv

(Assuming all other velocities are zero – during the contact, the ball is momentarily stationary)

So, Force = mΔv / t = (0.046 × 70) / 0.0005 = 6440N

Question 895: [Alternating Current > Rectification]

Alternating supply of frequency 50 Hz and having an output of 6.0 V r.m.s. is to be rectified so as to provide direct current for a resistor R. The circuit of Fig.1 is used.

The diode is ideal. The Y-plates of a cathode-ray oscilloscope (c.r.o.) are connected between points A and B.

(a)

(i) Calculate maximum potential difference across the diode during one cycle.

(ii) State potential difference across R when the diode has maximum potential difference across it. Give a reason for your answer.

(b) Y-plate sensitivity of the c.r.o. is set at 2.0 V cm^–1 and the time-base at 5.0 ms cm^–1.

On Fig.2, draw the waveform that is seen on the screen of the c.r.o.

(c) A capacitor of capacitance 180 μF is connected into the circuit to provide smoothing of the potential difference across the resistor R.

(i) On Fig.1, show the position of the capacitor in the circuit.

(ii) Calculate energy stored in the fully-charged capacitor.

(iii) During discharge, potential difference across the capacitor falls to 0.43V₀, where V₀ is the maximum potential difference across the capacitor.

Calculate fraction of the total energy that remains in the capacitor after the discharge.

Reference: Past Exam Paper – November 2006 Paper 4 Q6

Solution 895:

(a)

(i) Peak voltage = 6√2 = 8.48 V

(ii) The potential difference is zero because EITHER there is no current in circuit (and V = IR) OR all p.d. is across the diode

(b)

The waveform is that of a half-wave rectification.

{Peak voltage = 8.48V. 1cm represents 2.0V. 8.48V is represented by 8.48 / 2.0 = 4.24cm}

The peak height is at about 4.25 cm

{Frequency f is 50Hz. Period T = 1/f = 1/50 = 0.02s = 20ms. Half-period = 10ms. 1cm represents 5.0ms. 20ms is represented by 20 / 5 = 4cm. Half-period is represented by 2cm.}

The half-period spacing is 2.0 cm

(c)

(i) The capacitor should be shown in parallel with the resistor.

(ii)

EITHER Energy = ½ CV²      OR = ½ QV and Q = CV

Energy = ½ × (180×10^–6) × (6√2)² = 6.48 × 10^–3 J

(iii)

{EITHER Energy is proportional to V². Fraction = remaining energy / total energy.

Remaining energy is proportional to (0.43V₀)². Total energy is proportional to V₀².

So, fraction = (0.43V₀)² / V₀² = 0.43²

OR The actual value of the remaining energy could be calculated and the fraction could be found using the previously calculated value of the total energy in (c)(ii). But this is too time-consuming.}

EITHER fraction = 0.43²        OR final energy = 1.2 mJ

Fraction = 0.18