Please use euler euation. And use the way of external force equal to internal fo
ID: 3280673 • Letter: P
Question
Please use euler euation. And use the way of external force equal to internal force.
Problem 6-74: The uniform cylinder of mass 10 kg and radius 5 cm, as sketched in Figure P6-74, rolls without slip on a 30° inclined surface. A spring of spring constant 100 N/m is attached to a massless flexible inextensible strap that is wrapped around the cylinder. The spring is initially stretched 0.5 m. Sup- pose the system is released from rest with this initial stretch of the spring. Note that gravity acts on the system (a) Find the speed of the centroid of the cylin der and the angular speed of the cylinder when the spring is completely unstretched. 100 N/m 5 cm 10kg 30° > FIGURE P6-74 (b) Find the stretch of the spring when the speed of the cylinder becomes zero.Explanation / Answer
6-74. form the given data
mass of cylinder, m = 10 kg
rolls without slipping
radius R = 5 cm
angle of incline, theta = 30 deg
spring constant, k = 100 N/m
initial strech of spring, A = 0.5 m
a. initial PE of the system = 0
initial PE stored in the spring = 0.5kA^2 = 0.5*100*0.5^2 = 12.5 J
final PE stored in the spring = 0 J
Final PE of the system = mgAsin(theta)/2 = 10*9.81*0.5*sin(30) /2 = 12.2625 J ( this was divided by 2 becasue when the spring moves by distance dx, the center of mass of the cyloinder moves by distance dx/2)
hence KE of the cylinder = initial energy - final PE = 0.2375 J
KE of cylinder = 0.5mv^2 + 0.5Iw^2
here I is moment of inertia of cylinder, I = 0.5mR^2
v is linear speed of the cylinder
w is angular speed = v/R
hence
0.2375 = 0.5mv^2 + 0.25mv^2 = 0.75*10*v^2
v = 0.1779 m/s
angular speed = v/R = 3.559 rad/s
b. let strech of spring be x speed of cylinder is 0
then from conservation of eenrgy
12.5 = mg(A + x)sin(theta)/2 + 0.5kx^2 = 10*9.81*(0.5 + x)*sin(30)/2 + 0.5*100*x^2
50x^2 + 24.525x -0.2375 = 0
solving for x
x = 0.0095 m