Question

use newtons method

(1) :(20pts-each) Use the method of your choice to obtain the differential equations for the two systems given below. man lin No vM No Sup

for (a) assume m2>m1

Answer 1

Rightwards and upwards are +ve cordinates.

(a) For the block m1 the differential equation will be, (T is the tension on the string).

m1T1

For the second block we can write the differential equation as,

$m_2 \ddot{y_2} = T - m_2 g$

Subtracting the above two equations we get the combined differential equation as,

$m_1 \ddot{x_1} - m_2 \ddot{y_2} = m_2 g - kx_1 - cx_1$

(b) Here we will assume that there is no friction between the sphere and floor. Here there will be two equations, one regarding the rotation of sphere and one the displacement of center of mass of sphere. For displacement we get the equation as,

$m \ddot{x} = -kx - c \dot{x}$

For the rotation we have to take the moment of these forces, (clockwise is positive and anti clockwise is negative). Therefore we can write,

$I \ddot{\theta} = bc \dot{x} - akx$

The above two differential equations govern the system.