Question

A uniform solid disk of mass 3.4 kg and diameter 31 cm starts from rest and rolls without slipping down a 28 degree incline that is 7.75 m long.  g = 9.81 m/s^2 .

(a) Calculate the linear speed of the center of the disk when it reaches the bottom of the incline.

b) Determine the angular speed of the disk about its center at the bottom of the incline.

c) Through what angle (in radians) does this disk turn as it rolls down the incline?

d) Does the linear speed in (a) depend on the radius or mass of the disk?

1. The linear speed depends on the radius of the disk.

2.The linear speed depends on the mass of the disk.

3.The linear speed depends neither on the radius nor on the mass of the disk.

4.The linear speed depends both on the radius and the mass of the disk.

e) Does the linear speed in (b) depend on the radius or mass of the disk?

1. The linear speed depends on the radius of the disk.

2.The linear speed depends on the mass of the disk.

3.The linear speed depends neither on the radius nor on the mass of the disk.

4.The linear speed depends both on the radius and the mass of the disk.

Answer 1

A] By energy conservation,

initial PE = final rotational KE = Final translational KE

mgL sin theta = 0.5*iw^2+ 0.5mv^2

mgL sin 28 degree = 0.5*0.5mr^2*w^2 + 0.5mv^2 where w=v/r because of pure rolling

gL sin theta = 0.5*0.5v^2 +0.5v^2 = 0.75v^2

v = sqrt(gL sin theta/0.75)

= sqrt(9.81*7.75* sin 28 degree/0.75)

= 6.9 m/s answer

B] angular speed w = v/r = 6.9/(0.31/2) = 44.5 rad/s

C] angle = length/radius = 7.75/(0.31/2) = 50 radians

D] 3.The linear speed depends neither on the radius nor on the mass of the disk.

E] In part b, we didn't find linear speed but angular speed and Angular speed in (b) depends on radius of the disk.