This advanced mechanics problem involves finding the equation of motion of a sys

Question

This advanced mechanics problem involves finding the equation of motion of a system. Please be very detailed and explain all steps since I need help understanding this material. I will rate the answer postively if all work is shown.

Thank you.

a) b) Show that you can obtain the eqn of motion for the coordinate xby differentiating Econst. By applying Newton's second law to the masses and pully separately, show that the equation of motion is the same (you will then cancel out the two unknown tensions from the three eqn's) fr FT 2m24

Explanation / Answer

Let L be total length of string,

L = x' + pi R + x

Differentiating with respect to time,

0 = v' + 0 +v

again differentiating, 0 = a' + a

So velocities and accelerations of both blocks are equal in magnitude. Let acceleration of m1 be a downward. Let T1 and T2 be tensions in right and left respectively.

Now applying Newtons second law,

m1g - T1 = m1a...... 1

Again on second mass,

T2-m2g = m2a........2

Applying torque equation (Newtons second law of rotation) on pully,

(T1-T2)R = I*alpha

where I is moment of inertia of pully, alpha is its angular acceleration. If pully is disc, its moment of inertia will be 0.5 MR^2. Last equation becomes

T1-T2 = 0.5MR^2 alpha/R, and also alpha R =a,

  T1-T2 = 0.5Ma......3

Adding the three equations,

  m1g - m2g = (m1+m2 +0.5M)a

Acceleration a = (m1g-m2g) / (m1 +m2 +0.5M)

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