2. A differential equation relating the difference in tension \( T \), pulley contact angle \( \theta \) and coefficient of friction \( \mu \) is \( \frac{d T}{d \theta}=\mu T \). When \( \theta=0 \), \( T=150 \mathrm{~N} \), and \( \mu=0.30 \) as slipping starts. Determine the tension at the point of slipping when \( \theta=2 \) radians. Determine also the value of \( \theta \) when \( T \) is \( 300 \mathrm{~N} \). [273.3 N, 2.31 rads]
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An object of mass $m_{1}$ placed on an inclined plane (angle $\theta$ above the horizontal) is connected by a string going over a pulley to a hanging object of mass $m_{2}$ . Determine the acceleration of the system if the coefficient of static friction between object 1 and the surface of the inclined plane is $\mu_{s},$ and the coefficient of kinetic friction is $\mu_{\mathrm{k}}$ . If the problem has multiple answers, explore all of them.
(II) Suppose the force $F_{T}$ in the cord hanging from the pulley of Example 9 of "Rotational Motion," Fig. $21,$ is given by the relation $F_{T}=3.00 t-0.20 t^{2}$ (newtons) where $t$ is in seconds. If the pulley starts from rest, what is the linear speed of a point on its rim 8.0 s later? Ignore friction.
A mass of m1 = 50 kg is placed on an inclined plane with inclination angle θ = 30◦ and contact friction coefficient µ = 0.2. The mass m1 is connected through a massless string to a mass m2 = 20 kg which hangs in the air. The string goes over a pulley made from a uniform disk of mass M = 4 kg and radius R = 0.1 m. The disk rotates around a horizontal axis without friction and the string does not slip on the disk. The pulley’s angular acceleration is α = 50 rads/s2 . Ignore air resistance on each object.
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