In the following figure,$r_1 = 5 \, cm$ and $r_2 = 30 \, cm$. If the moment of inertia of the wheel is $5100 \, kg \cdot m^2$,then its angular acceleration will be:

$A$ crew of scientists has built a new space station. The space station is shaped like a wheel of radius $R,$ with essentially all its mass $M$ at the rim. When the crew arrives,the station will be set rotating at a rate that causes an object at the rim to have radial acceleration $g,$ thereby simulating Earth's surface gravity. This is accomplished by two small rockets,each with thrust $T$ newtons,mounted on the station's rim. How long a time $t$ does one need to fire the rockets to achieve the desired condition?

$A$ rectangular solid rod of length $0.3\, m$ is held horizontally,with one of its sides on the edge of a platform of height $5\, m$. When released,it slips off the table in a very short time $\Delta t = 0.01\, s$,remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to:

$A$ wheel is rotating at $600$ rpm about its axis of rotation. When the power is cut off, it comes to rest in half a minute. Its angular retardation expressed in rad/s$^2$ is (retardation is uniform).

$A$ massless rod of length $2l$ has equal point masses $m$ attached to its two ends as shown in the figure. The rod is rotating about an axis passing through its centre and making an angle $\alpha$ with the axis. The magnitude of the rate of change of angular momentum of the rod,i.e.,$\left|\frac{dL}{dt}\right|$,is equal to:

$A$ solid disc has a diameter of $0.5 \ m$ and a mass of $16 \ kg$. What torque must be applied to increase its angular velocity from $0$ to $120 \ rpm$ in $8 \ s$?