You are investigating two mysterious objects with massM = 0.7 kg and diameter R = 0.2 m, which arerolling down a slope without slipping. The slope makes an angle θ =30 ° with the horizontal. The acceleration of gravity for an objectin free fall is 9.8 m/s².

(a) Draw a force diagram, showing all the forces on theobject.

[2 marks]

(b) You are measuring the time it takes the objects, from restand without slipping, to roll down the L = 1.00 m longslope, as measured along the slope. Show that the arrival time atthe bottom for an object with moment of inertia I =cMR² is

t = 2 ( 1 + c ) L g sin ⁡ θ

[3 marks]

(c) Like a good experimentalist, you are taking a number ofrepeat measurements. In the following, make sure you quote allanswers to an appropriate precision. Your measurements are capturedin the following table.

Take	Object 1 (s)	Object 2 (s)
1	0.8	0.93
2	0.74	0.93
3	0.82	0.91
4	0.77	0.8
5	0.83	0.95

          (i) Findthe average times that the two objects take to roll down the slope.Include the associated uncertainties in your result.

[2 marks]

          (ii)Deduce the accelerations of the two objects, with uncertainties.Show your working.

[2 marks]

          (iii)Explain why the two objects roll with different accelerations.

[2 marks]

          (iv) Whatis your best guess of the geometry of your objects? Providereasoning for your guess.

[2 marks]