Studying for my final exam, these are the previous tests, I need help understand

Question

Studying for my final exam, these are the previous tests, I need help understanding them completely. If you may

please answer with formulas used, and any thoughts that might have been used to answers these and a clear

explanation of the process used to achieve the answer.

As shown in the figure, a glider of mass m1 (10.0 kg) can slide without friction on a horizontal air track. It is attached to an object of mass (5.0 kg) by a massless string. The pulley has radius R (10.0 cm) and moment of inertia I (0.05 kg m2) about its frictionless axis of rotation. When released, the hanging object accelerates downward, the glider accelerates to the right, and siring turns the pulley without slipping or stretching. Using Newton's second law and rotational analog of Newton's second law for a rigid body, calculate the angular acceleration a of the pulley (Hint: write down equations for mi, m2, and pulley, respectively.)

Explanation / Answer

since the pulley doesnt stretch, horizontal acceleration of m1 = vertical acceleration of m2.

acceleration of m1 = T1/m1

acceleration of m2 = g-T2/m2

equating these two .. T1/10 = 10-T2/5 => T1+2T2 = 100

angular acceleration of the pulley is given by T = I?

T = (T2-T1)R (R = 0.1 m)

I = 0.05 kgm^2

? = (T2-T1)*0.1/0.05 = 2(T2-T1) rad/s^2

since the string doesnt slip, ?R = a

=> 2(T2-T1) * 0.1 = T1/10

=> 2T2 = 3T1

=>T2 = 3/2T1

applying this in the equation T1+2T2 = 100

we get 4T1 = 100

T1 = 25N

T2 = 37.5N

angular acceleration ? = 2(37.5-25) = 25 rad/s^2

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