Trying to study for my physics exam any help with the justif

Trying to study for my physics exam, any help with the justification for these problems would be greatly appreciated! (showing 10-14)

10. A solid, uniform sphere of mass 2.0 kg and radius 1.7 m rolls without slipping down arn inclined plane of height 7.0 m. What is the angular velocity of the sphere at the bottomm of the inclined plane? 5.8 rad/s 11. A solid disk of radius 1.60 m and mass 2.30 kg rolls without slipping to the bottom of an inclined plane. If the angular velocity of the disk is 5.35 rad/s at the bottom, what is the height of the inclined plane? 5.61 m 12. A steel guitar string with a diameter of 0.300 mm and a length of 70.0 cm is stretched by 0.500 mm while being tuned. How much force is needed to stretch the string by this amount? (Y-2.0x1011 Pa) 10.1 N 13. An irregularly shaped object 10 m long is placed with each end on a scale. If the scale on the right reads 74 N and the scale on the left reads 93 N, how far from the left is the center of gravity? 4.4 m 14. A roller coaster, loaded with passengers, has a mass of 2000 kg; the radius of curvature of the track at the bottom point of the dip is 24 m. If the vehicle has a speed of 18 m/s at this point, what force is exerted on the vehicle by the track? 4.7x104 N

Solution

13) x = L N₂ / N₁ + N₂

= 10 * 74 / (74 + 93 )

= 4.4 m

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14) a = v² / r = 18² / 24 = 13.5 m/s²

F_net = m g = 2000 * 13.5 = 27000 N

F_grav = m g = 2000 * 9.8 = 19600 N

F_norm = 27000 + 19600 = 46600

= 4.7 * 10⁴ N