Question
Two small balls $A$ and $B$, each of mass $m$, are joined rigidly by a light horizontal rod of length $L$. The rod is clamped at the centre in such a way that it can rotate freely about a vertical axis through its centre. The system is rotated with an angular speed $\omega$ about the axis. A particle $P$ of mass $m$ kept at rest sticks to the ball $A$ as the ball collides with it. Find the new angular speed of the rod.
Step 1
The moment of inertia for the initial condition, $I_1$, is given by the sum of the moments of inertia of the two balls about the axis of rotation. Since the balls are at a distance of $L/2$ from the axis of rotation, we have: \[I_1 = m Show more…
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Two small balls $A$ and $B$, each of mass $m$, are joined rigidly to the ends of a light rod of lengh $L$ (figure 10-E10). The system translates on a frictionless horizontal surface with a velocity $v_{0}$ in a direction perpendicular to the rod. A particle $P$ of mass $m$ kept at rest on the surface sticks to the ball $A$ as the ball collides with it. Find (a) the linear speeds of the balls $A$ and $B$ after the collision, (b) the velocity of the centre of mass $C$ of the system $A+B+P$ and $(c)$ the angular speed of the system about $C$ after the collision. [Hint : The light rod will exert a force on the ball $B$ only along its length.]
Two small balls with mass M and m are connected by a massless, rigid rod of length L. This assembly is rotated around an axis perpendicular to the rod, and at a distance x from the mass M. Note: you may assume that the axis of rotation is located between the two masses. a. Calculate the moment of inertia about this axis. b. Show that if the axis of rotation passes through the center of mass, then this moment of inertia is I = μL^2, where μ = mM/(M+m); μ is known as the "reduced mass" of the system. c. If both balls are solid spheres of radius r < L/2, what is the moment of inertia about the center of mass.
Two small balls with mass M and m are connected by a massless, rigid rod of length L. This assembly is rotated around an axis perpendicular to the rod, and at a distance x from the mass M. Note: you may assume that the axis of rotation is located between the two masses. a. Calculate the moment of inertia about this axis, b. Show that the moment of inertia is minimized when the axis passes through the center of mass, and c. Show that this moment of inertia is I = ̀́L², where ̀́ = mM/(M+m); ̀́ is known as the "reduced mass" of the system.
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