Conservation of Angular Momentum: Videos & Practice Problems

A researcher studying rotational inertia and angular velocity constructs an object to simulate the human body and the effects of arms and hands. The object has a central trunk and rods representing arms/hands. When the rods are outstretched, they are treated like thin rods rotating about an axis centered on the trunk. When the rods are lowered, they are treated like hollow cylinders. The rods representing arms/hands have a total mass of 10.2 kg, each 0.750 m long. When lowered, they become a cylinder of radius 30.0 cm. The trunk has a constant moment of inertia equal to 8.10 kg•m². If the object has an initial speed of 1.20 rad/s, determine its speed when the rods are lowered. 

A child's toy has a cube that rotates on a level board with negligible friction. The cube has a mass of 0.350 kg and is rotating on one end of a string of negligible mass passing through a hollow at the center of the board. The cube is rotating at 3.20 rad/s and is located at a distance of 0.450 m from the hollow when the child reduces the cube's radius of rotation to 0.220 m by pulling the string from beneath the board. Calculate the work done by the child in shortening the string. Treat the cube like a point mass. 

A block of mass 0.0200 kg is rotating on a flat bench with negligible friction. It is held in the circle by a massless string running through a hollow in the bench. At first, the block is rotating at a radius of 0.350 m from the hollow at an angular speed of 1.95 rad/s. The string is shortened by pulling from below the bench, reducing the rotational radius to 0.180 m. Determine the change in kinetic energy of the block. Treat the block like a particle. 

During an experiment, the wheel of a laboratory gyroscope precesses in a horizontal plane at 30 revolutions per minute. The wheel (m_wheel = 200 g) that is fixed to a frame is free to spin about the shaft (m_shaft = 25 g) of the frame. The moment of inertia of the wheel about the shaft is 10^–4 kg•m²  The gyroscope is supported by a pivot near the right-hand end of the shaft located 5 cm away from its center of gravity, as shown in the figure. Determine the force exerted by the pivot on the shaft.

A spacecraft traveling from Earth to Mars uses a gyroscope to maintain its orientation and angular velocity. What will be the precession rate of the gyroscope near Mars's surface If the precession rate near Earth's surface was 1 rad/s? The surface gravity on Mars is 38% of the surface gravity on Earth.

A uniform rod of length 0.80 m is fixed to a massless axle on one of its ends such that its rotational axis is perpendicular to its length. The rod's moment of inertia about the axle is 0.0410 kg•m². The rod is rotating at 1.40 rads/s when a remote-controlled movable mass slides from very close to the axle to the opposite end of the rod. The mass has a linear speed of 0.980 m/s when located at the opposite end. Determine the mass of the rod. You may treat the mass as a particle. 

Neutron stars are believed to form when a massive star depletes its fuel and collapses. The collapse crushes together all protons and electrons into neutrons. The density of a neutron star is about 10¹⁴ times greater than the density of the sun. Imagine that the sun (with a radius of 6.96 × 10⁵ km) collapses into a neutron star of radius 15 km. Using the sun's average rotational speed of one rotation every 27 days, what would be the rotational speed of the neutron star formed? Model the sun and the neutron star formed as uniform solid spheres. 

A family dining table has two circular wooden table tops. A small top spins about a vertical axis through its center on top of a large and stationary tabletop below it. Meals are taken from the larger top while Foods and drinks are circulated from the rotating smaller top. Suppose the smaller top has a mass of 11.4 kg and a radius of 2.00 m and is rotating at 2.30 rad/s when a pot of mass of 4.50 kg is placed near its outer edge. Determine the system's kinetic energy before and after the pot is placed down. Treat the small top like a disk and the pot as a point mass.

A wooden dining table has two concentric circular surfaces. People feed on the outer stationary surface. Foods and drinks are placed on the inner surface (radius 1.20 m and mass 6.75 kg) and then circulated to members on the dining table. The inner surface is rotating at 2.50 rad/s about its center on a vertical axis when a dish of mass 5.00 kg is placed on it gently at a point very close to its outer edge. Calculate the angular speed of the inner surface once the dish has been placed on it. You may treat the dish as a point mass.

A double-sided tape covers the circular surface of a plate rotating about its central axis. The plate has a mass of 750 g and a radius of 7.5 cm. An experimental setup is designed in such a way that two spheres of mass 125 g each are dropped simultaneously from a height h and stick to the plate. The location of impact is precisely controlled at the opposite ends of a diameter. The plate spins initially at 185 rpm. Find the plate's rotational speed after the impact.

A student throws a 5.0 g tip in the x-direction with a speed of 25 m/s toward a square-shaped, vertical swinging plate made of foam, attached to a pivot at one end. The plate has a mass of 125g, and a side of 80 cm. The tip is implanted in the foam 5 cm from the opposite side of the pivot. Calculate the angular speed of the plate just after the collision.  

An exoplanet moves in an elliptical orbit around its Sun, as depicted in the diagram. The two planets are attracted to each other by gravitational force. The speed of the planet at the vertex (v₁) of its trajectory is 800 km/h. Calculate the speed of the planet at the second vertex (v₂).

A 10 kg fox terrier dog runs and jumps through a hoop onto a rotating circular platform in a circus scene. The platform, with a mass of 65 kg and a radius of 1.25 m, rotates at 43 rpm around its frictionless central axis. The dog is trained to run at a speed of 2.0 m/s in the same direction as the platform's rotation and tangent to the platform. The dog lands exactly on the platform's edge. Determine the angular velocity of the platform after the dog has landed on it.

When a massive star exhausts its fuel and collapses, neutron stars form. Pulsars are the most common type of neutron star. Pulsars are rotating neutron stars that emit radiation pulses at very regular intervals, typically once per rotation. Astrophysicists measure rotation rates by detecting electromagnetic waves emitted by the magnetic field's poles. Consider a pulsar, formed after the gravitational collapse of a massive star and emitting electromagnetic waves once every half second. The pulsar is formed from a massive star that has a mass of M = 1.8 × 10²⁷ kg, a diameter of d, and a period of revolution of 640 hours. The radius of the pulsar is  2.1 × 10⁴ m. Calculate the diameter of the original massive star. Consider the star and pulsar to be rigid spherical objects having the same mass.

Initially a physics professor is sitting on a rotating chair with his arms extended laterally. The rotation speed of the chair is 3.0 revolutions every 2.0 s while his arms are outstretched. Note that the moment of inertia for the professor is 5.2 kg.m². Later on in the experiment, the professor increases his rotational speed by himself to 4.5 rev/s without using his feet to accomplish this result. Determine what his final moment of inertia value will be considering this increase in rotational speed. Also, please explain how this increase in rotational speed accomplished?

A 47 kg monkey is standing at the center of a disk that is rotating horizontally at an angular velocity of 0.72 rad/s. The moment of inertia of the disk is 840 kg.m² and its radius is 2.0 m. The monkey walks straight to the edge of the disk while the disk rotates without any external forces being applied to it. Given that there is no friction, determine the disk's angular velocity after the monkey reaches the edge.

A 42 kg dog is at the center of a large disk of radius 6.0 m. Initially, the disk rotates horizontally with an angular velocity of 0.82 rad/s about a vertical axis through its center. Afterward, the dog walks directly to the edge of the disk. Given that the moment of inertia of the disk is 870 kg·m², determine the total rotational kinetic energy of the system with the dog initially on the disk at the center first and then do the same for when the dog walks to the edge of the disk. Assume that the disk rotates without any friction.

A merry-go-round is rotating at a rotational speed of 6.2 rev/s. A beam at rest, of mass equal to that of the merry-go-round and length equal to the diameter of it, is placed onto the spinning platform. The beam is placed in such a way that the center of it coincides with the center of the merry-go-round. Afterward, the merry-go-round and the beam spins together. Calculate the final rotational speed of the system. (Assume that the merry-go-round is a uniform thin cylinder that rotates without any friction about its axis, and the beam only has length.)

At t = 0, a merry-go-round of mass m and radius r rotates horizontally at an angular velocity of ω_i about a perpendicular axis through its center. A child at the edge of it begins moving toward the center in a straight line at the same time with a constant speed of v_o with respect to the merry-go-round. Given that the mass of the child is M, express the angular velocity of the merry-go-round as a function of time assuming that there is no friction and that the merry-go-round is a thin cylinder.

In a circus, a large horizontal disk of mass m and radius r rotates at an angular velocity of ω_i about its vertical axis through its center at t = 0. An acrobat of mass M starts walking toward the center of the disk from his initial position at the edge at the same time. If the speed with which the acrobat walks toward the center is v_o with respect to the disk, determine the angular velocity of the disk-acrobat system after he reaches the center. (Assume that the disk rotates without any friction.)

Consider a massive asteroid, initially shaped like a perfect sphere, with a radius of 155 km and mass of 1.05×10¹⁵ kg, spinning at a rate of one revolution every 24 hours. Suppose this asteroid were to fragment into smaller pieces, expelling 60.0% of its mass in the process but conserving angular momentum, and the largest remaining fragment, which remains spherical, has a radius of 20.0 km. What would be the new rotational speed of this fragment?

Imagine a spherical star with the same mass as Jupiter but half its radius, rotating at a speed of 1 revolution every 7 days. If it were to collapse into smaller segments, losing 80% of its mass in the process but conserving its proportional share (80%) of the initial angular momentum, and assuming it remains a uniform sphere, with the remaining segment having a radius of 1/500 of the original what would be the new rotational speed of the resulting segment?

Imagine a circus performer spinning on a giant rotating platform. The performer's body can be approximated as a solid cylinder with a mass of 60.0 kg, a radius of 0.4 m, and a height of 1.8 m. Each arm can be approximated as a thin rod with a mass of 3.0 kg and a length of 0.7 m. If the performer stands on the platform with arms outstretched and completes one rotation in 2.5 s, what would be the new time for each rotation if the performer brings their arms close to their body? Ignore any frictional effects and assume the platform is weightless.

A gymnast is positioned on a rotating platform with arms extended outward. We approximate the gymnast's body as a cylinder with a mass of 52.0 kg, a radius of 0.25 m, and a height of 1.80 m. Each of the gymnast's arms, treated as thin rods, weighs 2.5 kg and has a length of 0.40 m. When the gymnast's arms are fully extended horizontally, the rotation period of the platform is measured to be 1.5 s. Subsequently, the gymnast retracts their arms toward their body. Calculate the change in kinetic energy as the gymnast transitions from the arms-down to the arms-extended position. Assume the moment of inertia of the platform is negligible.

A daredevil performs on a rotating stunt platform with his arms outstretched. The moment of inertia with his arms outstretched is 3.5 kg.m² and with his arms closed to his sides is 2.7 kg.m². The stunt platform completes one rotation in 2.2 s when the daredevil's arms are fully extended and 1.7 s when the arms are closed. By calculating the change in the kinetic energy during the transitions from arms-down to arms-outstretched position, state whether it is easier for the daredevil to raise their arms while the platform is rotating or while it is stationary.

Two circus performers, each weighing 60 kg, ride toward one another on individual unicycles along the parallel paths spaced 1.0 m apart. Both are moving at a speed of 2.5 m/s with their arms outstretched. Upon meeting, they grasp each other's hands and continue cycling, preserving the initial separation distance. Consequently, they commence rotating around a shared center. If they subsequently tug on each other's hands, decreasing their separation to 0.7 m, what will be their mutual angular speed after this alteration? Ignore friction and mass of the unicycle.

A bowling ball with a radius of $r$ is rolled down a wooden lane. It starts from a stationary position at point A and is given an initial linear speed of $v_0$ and an angular speed of $\omega_0$ that spins it backward (i.e., reverse spin when viewed from the side). As the ball skids, it encounters kinetic friction. Using conservation of angular momentum, determine the critical angular speed $ω_C$ such that, if $\omega_0$ = $ω_C$, kinetic friction will not only slow the ball temporarily but will bring it to a complete stop.

A solid sphere with a radius $r$ is rolled on a rough concrete surface. It starts from a stationary position at point A and is given an initial linear speed v₀ and an angular speed ω₀ that imparts a reverse spin on the ball. As the ball skids, kinetic friction acts upon it. If ω₀ is 20% smaller than the critical angular speed $\omega_{C}$, i .e., $\omega_0{}=0.80\omega_{C}$, determine the ball's center-of-mass velocity $v_{CM}$ at the moment it starts to roll without slipping. The critical angular velocity, $\omega _{\mathrm{C}}$, is reached when kinetic friction causes the ball to come to a complete stop rather than just momentarily, meaning $\omega _0=\omega _C$.

Hint: The ball possesses two types of angular momentum: the first due to the linear speed $v_{CM}$ of its center of mass (CM) relative to point A, and the second due to the spin at angular velocity ω about its own CM. The ball’s total angular momentum L about point A is the sum of these two angular momenta.

A marble sphere with radius r rolls on a smooth floor from point A. Initially at rest, it starts with a linear speed $v_0$ and a reverse angular speed $\omega_0$, which is 15% higher than the critical angular speed $\omega _{\mathrm{C}}$( $\omega_0=1.15\omega_{C}$), causing skidding. Calculate the velocity $v_{CM}$ of the sphere's center of mass when it starts rolling smoothly without slipping. Remember, $\omega _{\mathrm{C}}$ is the angular velocity at which the sphere stops due to kinetic friction. The sphere's total angular momentum $L$ about point $O$ includes contributions from both its center of mass linear speed $v_{CM}$ and spin angular velocity ω.

Two cyclists, both weighing 70 kg, travel along parallel paths separated by 2.0 meters. Both cyclists are moving at a speed of 3.0 m/s with their arms extended. As they pass each other, they connect by holding a rope, maintaining the 2.0-meter separation, and start rotating around a common center. Treating the cyclists as point masses regarding their rotational inertia, calculate the change in kinetic energy when they pull on the rope, reducing their radius to half its original value. Ignore friction and the mass of the bicycles.

A 3.4-meter-diameter disk spins at 0.90 rad/s with a moment of inertia of 1740 kg·m². What would be the effect on the disk if five cats were initially on the disk and jumped off radially?

A child is playing with a tiny marble ball of mass 0.055 kg on a frictionless, horizontal table. She attaches it to a massless string and passes the string through an opening in the table. Initially, she spins the marble ball in a circle about the opening with a radius of 0.600 m and with an angular speed of 5.0 rad/s. Afterward, she pulls the string from below, reducing the circle's radius in which the marble ball revolves to a shorter one of 0.300 m. Calculate the new angular speed. Assume that the marble ball is a point particle.

A child weighing around 40 kg stands on the rim of an idle carousel that has a diameter of 10 m and a moment of inertia of 2000 kg·m². Note that the carousel can rotate without any significant friction. When she starts running along its circumference at a speed of 3.0 m/s, the carousel will begin to move in reverse direction. What will be its angular velocity in this case? Please treat the angular velocity of the child with respect to the ground to be positive and think of the carousel as a solid cylinder.