Question

A 10 kg disk is released from rest and is free to either roll or slide down the incline toward a spring. The coefficient of friction between the disk and surface is 0.7 and 0.9. The radius of the disk is 1 m. The spring stiffness is 5 N/m. Answer the following questions about the system shown in the figure below. a) Will the disk roll or slide down the incline? Support your answer mathematically. b) What is the velocity of the disk's center of gravity when in reaches the bottom of the incline? c) What is the maximum deflection of the spring? 10 m 30° w

Answer 1

a "mgsine mgcoso ng Us=0.9 luk -0.7 N> Normal force f = frictional force = MgN Force = ma & fx = max mgsino - fe max Now, 2 - Torque - Ix - FXR 2= Ox R dhe where R = Radius of disc I ax = FXR R

F = I ak R2 Noco, I > _MR2 fz IMR² an २ R2 f = Iman 2 the value ax Comparing case of sliding slipping 0 and ② are not equal therefore this is or rolling with a b) Let Vem be the velocity of center of gravity of of disc.

[kE+ PE [kE+ PE intal final + thw2 mg h= 1 m vent 2 6= angular velocity me=n cm wa Vam ४ 2 Vem I tur WT mgh = Lam 5 (3m22) R² migh. Im Vam ut I m vom? 4 2 2 3 vom ghz 4. Vom | 4gh 4 x 9.81810 3 3 Vem 11.43 mis

transfered The energy of the disc will be ed to the spring righ = ļ kx2 k = 5 W/m 22 deflection of spring 10 X 9.81 x 10 5x5x0? x 19.8m

a) Disc will slide

b) Velocity of centre of mass is 11.43 m/s

c) maxximum deflection of spring 19.8 m