Rigid body dynamics

Four point masses of $0.50\mathrm{kg}$ each are fixed at the vertices of a square frame of side $0.40\mathrm{m}$ and negligible mass. Compute the moment of inertia of the system about the axis perpendicular to the plane of the square through the center and about the parallel axis through a vertex.

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A uniform thin ring of mass $0.50\mathrm{kg}$ and radius $0.30\mathrm{m}$ rotates about an axis perpendicular to the plane of the ring through a point of the ring. Compute the moment of inertia about this axis.

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A wheel with moment of inertia $2.5\mathrm{kg}\cdot {\mathrm{m}}^2$ about its own axis turns at $90\text{rpm}$. A brake applies a constant torque of $3.0\mathrm{N}\cdot \mathrm{m}$ that opposes the rotation; the friction of the axle is negligible. Compute the stopping time and the number of revolutions made during braking.

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A uniform solid cylinder of mass $8.0\mathrm{kg}$ and radius $0.20\mathrm{m}$ rotates without friction about its own horizontal axis. An inextensible string of negligible mass, wound around the cylinder without slipping, supports a block of mass $1.0\mathrm{kg}$. Compute the acceleration of the block and the tension of the string.

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Two blocks of masses $5.0\mathrm{kg}$ and $3.0\mathrm{kg}$ hang from the ends of an inextensible string of negligible mass that passes, without slipping, over a pulley in the shape of a uniform disk of mass $2.0\mathrm{kg}$ and radius $0.10\mathrm{m}$, free to rotate without friction. Compute the acceleration of the blocks and the tensions in the two segments of string.

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A uniform rod $0.90\mathrm{m}$ long, hinged without friction at one end, is released from rest in a horizontal position. Compute the angular velocity of the rod and the speed of the free end when the rod passes through the vertical.

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A motor applies a constant torque of $12\mathrm{N}\cdot \mathrm{m}$ to a wheel, initially at rest, with moment of inertia $0.80\mathrm{kg}\cdot {\mathrm{m}}^2$ about the axis; friction is negligible. Compute, after $4.0\mathrm{s}$, the angular velocity, the kinetic energy of the wheel and the power delivered by the motor at that instant.

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A uniform solid cylinder starts from rest and rolls without slipping down an inclined plane, descending through a height difference of $0.80\mathrm{m}$. Compute the velocity of the center of the cylinder at the end of the descent.

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A uniform solid sphere descends along a plane inclined at $40^{\circ }$. Compute the minimum coefficient of static friction between sphere and plane for which the motion is pure rolling.

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A platform in the shape of a uniform disk, of mass $150\mathrm{kg}$ and radius $1.5\mathrm{m}$, rotates without friction about its own vertical axis at $0.80rad/s$. A person of mass $75\mathrm{kg}$, with zero initial angular momentum about the axis, jumps onto the rim of the platform and stays there at rest relative to it. Treating the person as a point mass, compute the final angular velocity.

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A uniform rod of mass $1.5\mathrm{kg}$ and length $1.2\mathrm{m}$ hangs at rest from a frictionless hinge located at its upper end. A bullet of mass $10\mathrm{g}$, traveling horizontally at $300m/s$ perpendicular to the axis of the hinge, strikes the lower end of the rod and stays embedded in it. Compute the angular velocity of the rod immediately after the collision.

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A uniform beam of mass $80\mathrm{kg}$ and length $6.0\mathrm{m}$ rests horizontally on two supports placed at the ends $A$ and $B$. A person of mass $70\mathrm{kg}$ stands on the beam $1.5\mathrm{m}$ from $A$. Compute the reactions of the two supports.

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A uniform ladder leans against a smooth vertical wall; the coefficient of static friction between the ladder and the floor is $0.35$. Compute the minimum angle that the ladder can form with the floor while remaining in equilibrium.

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