15. (15 points) A thin spherical shell of radius R = 5.0 cm and mass M = 2.0 kg is placed at the top of an incline. The surface of the incline makes an angle θ = 25° with the horizontal. The shell is released from rest and rolls without slipping. [Icom = 2/3MR² for spherical shell]
a) What is the acceleration of its center of mass (acom) during rolling?
b) What is the magnitude and direction of the force of friction required to maintain pure rolling?
c) If it reaches an angular speed of ω = 84 rad/s at the bottom of the incline, what is the height h of the incline?
Answers: a) 2.48 m/s² ; b) 3.31N, directed up the incline; c) 1.5m
16. (15 points) Figure shows a sports car capable of reaching 100 km/h speed in under 3 seconds (t < 3s). During such an acceleration, an average force of magnitude 5823 N is applied on the front wheel from its center of mass (assuming constant). The tire carries an average mass of 300 kg and the rotational inertia of the wheel is 20.82 kg.m². The tire radius is 32 cm. If the wheels roll smoothly during the motion, find
a) the linear and angular acceleration of the wheel,
b) the direction and magnitude of the frictional force on the wheel.
Answers: a) a =11.57 m/s², α =36.15 rad/s²; b) fs=2352.4 N, rightward/ opposite to the direction of motion
17. (15 points) Three point particles are attached to a disc which is initially rotating with wi = 2.4 rad/s at t = 0 s around its center. The masses of the objects and distance between them are m1 = 1.5 kg, m2 = 1.8 kg, m3 = 2.0 kg and d = 3 cm. If the disc accelerates at a rate of 0.6 rad/s²;
a) Find linear speeds of masses after 3 s.
b) How many revolutions does m1 make in first 3 s?
c) What is the kinetic energy of the system at t = 2 s? (Icom,disc= 4.8 kgm²).