Suppose a drum is a uniform cylinder with a mass of 12.0 kg and a radius of 32.0 cm, mounted on a wall with the axis of rotation perpendicular to the wall, rotating on frictionless bearings. A bucket is suspended from the free end of a rope wound around the outer rim of a drum. Determine the mass of the bucket if it travels 13.4 m in the first 4.00 s from rest.

A child's wooden toy has the form of a sphere. The child throws the ball so that it rolls down a surface that is inclined at an angle θ above the horizontal without slipping. If the ball behaves like a uniform solid sphere, determine the acceleration of the ball's center of mass.

A small object attached to a thread is being swung around counterclockwise in a circle of radius R_o with uniform angular acceleration α. The position vector r of the object is given by: r̄ = î R_o cos Φ + ĵ R_o sin Φ where Φ = ω_it + (1/2)αt², here ω_i denotes the initial angular velocity and t is the time elapsed. Given that the mass of the object is m, and its moment of inertia is I, find its tangential acceleration a_t, and using τ̅ = Iα̅, find the torque acting on it.

A 55 kg athlete preparing to dive stands at the end of a homogenous horizontal diving board of mass 38 kg. The diving board is 6.0 m long. The other end is anchored through a screw to the diving tower. Calculate the magnitude of the torque exerted by the athlete and the board about the screw.  

To open a door, you exert a force at a point with a position vector r= 2 (m) i - 3 (m) j with respect to the hinge. The exerted force is given in unit-vector notation as F= 15 N i - 10 N j. The hinge is located at the origin of the Cartesian plane. In a graphic, draw the position vector r, the force F, and the origin.

A very light bar of length 0.80 m has two cubes of mass 0.25 kg and 0.75 kg welded to its two ends. The bar attached at its center of mass to a fan motor revolves counterclockwise at a steady angular speed of 45 rpm. A torque τ applied to the bar causes it to stop rotating in 12 s. Calculate τ.

A building employs a dual counterweight system with two elevator cars. Elevator Car A has a mass of 1200 kg and starts at the tenth floor. Elevator Car B, with a mass of 1250 kg, begins at the second floor. They are connected by a steel cable that runs over a pulley that is free to rotate about a fixed axis at the top of the shaft. The pulley is a solid cylinder with a radius of 0.80 meters and a mass of 100 kg. Find the acceleration of each elevator car as they move in the shaft.

A 35 kg dog begins to cautiously tread on a 6.2 meter long, 58 kg wooden plank that is supported by two support beams. With safety in mind, determine the nearest distance the dog can approach the right end of the wooden plank without causing it to topple over.

An 8.0×10⁴ kg aircraft is orbiting Earth at a constant velocity. The forces acting on it are depicted in the figure below. The thrust generated by the onboard propulsion system is F_w = 7.2×10⁵N, acting on a line 1.8 m below the aircraft's center of mass. Calculate the drag force $F_D$ exerted by the atmosphere and the distance above the center of mass where it acts. Assume that both F_D and F_W are horizontal. (F_lift represents the lift force generated by the aircraft's wings, if any.)

A uniform beam of mass $m$ and length $L$ leans against a rough wall as shown below. The coefficients of static friction between the beam and the floor, and between the beam and the wall, are $\mu_{f}=0.50$ and $\mu_{w}=0.30$ , respectively. Assume the beam is on the verge of slipping and stability is ensured if $\theta \geq \theta_{\min }$, given by

Calculate the true value of $\theta_{\min }$. Then, assuming the wall is frictionless ($\mu_{w}=0$), the expected value of $\theta _{\min }\text{ is }45^{\circ }$. Determine the percentage error between the true value and the frictionless wall approximation.

A traffic light of mass 3 kg hangs from the right end of a nonuniform bar as shown in the image. The non-uniform bar is 7 m long and has a mass of 10 kg. The center of gravity of the bar is 4.00 m from its left end. The bar is held by a frictionless pivot at its left end and a light cord at a distance of 2.00 m from the pivot. The light cord is perpendicular to the bar. In its equilibrium position, the bar makes an angle of 30.0° below the horizontal. Determine i) the tension (T) in the cord as well as ii) the force (P) exerted by the pivot on the bar. A free-body diagram of the bar will be very helpful.

A uniform 87-kg column is held still by a cable that is attached to it at a point that is 3/4 of its length from the ground. The other end of the cable is attached to the ground which makes an angle of 33° with the horizontal. Given that the length of the column is 16 m and the angle between the column and the ground is 83°, calculate the tension in the cable.

A clamp is designed to secure a lightweight metal tube. When both the finger and the thumb are applied with equal forces of F_F=F_T=12.0 N, calculate the resulting force that the clamp's ends exert on the metal tube.

A diver spins in mid-air during a dive. The diver's body can be modeled as a solid cylinder which is ≈ 0.65 M including the diver's head, whereas his arms can be approximated and treated as thin rods equal to 0.20 M for both individual arms, noting that M is the total mass of the diver's body. Calculate how much faster the diver spins when he pulls in his arms towards his body. Assume that the length of each arm "r" is twice the value of the radius "R" of the cylindrical body.

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 cylindrical ring with a mass M and a radius R has a delicate cord coiled around it. One end of the cord is secured while the ring descends vertically from a stationary position, uncoiling the cord as it goes. Calculate the angular momentum of the ring about its center of mass as it varies with time.

A uniform window that is 0.700 m long and 0.500 m wide has a mass of 11.0 kg. The window is pivoted by frictionless hinges along its width and allowed to hang vertically. A 0.850 kg unlucky peregrine falcon has a level flight speed of 100 km/h when it hit the window at its center. The falcon bounces back at a speed of 60 km/h. Calculate the window's angular speed immediately after its hit by the unlucky falcon.

One end of a stationary rod is connected to a pivot on a smooth horizontal surface. The rod can rotate freely without friction. On the other end, a 1500.0-g squirrel stands still. Given that the rod has a mass of 75.0 g and it is 120 cm long, calculate the angular speed of the rod if the squirrel jumps off in the horizontal direction, perpendicular to the rod, with a speed of 25.0 cm/s relative to the surface, just after the squirrel jumps. Assume that the rod has a uniform mass distribution.

A tiny marble with a mass of 15 grams is moving in a frictionless circular path with a radius of 50 cm. The marble moves at an initial speed of 4 rad/sec. The radius then decreases to half its original size. Is the angular momentum of the marble conserved? Explain your answer.

A motor-driven floor scrubber has a mass of 1.5 kg and a diameter of 430 mm. The scrubber rotates about an axle attached to its center. The scrubber sweeps an angle as a function of time given by θ(t) = At² + Bt⁴. If θ is in radians, t in seconds, and A and B are equal to 0.820 and 0.710, respectively, determine the units of A and B.

An observer is at point o. A car of mass m traveling at a constant velocity u is about to pass by her from the left as shown in the figure. Treating the car as a particle, determine the angular momentum of the car about the point o.

In an engineering lab, a balancing wheel has two 40 cm rods, each with a 450 g mass at its end, mounted 180° apart on a central axle. The rods are spaced 60 cm apart along the axle, and the setup spins at 4.0 rad/s.

i. Calculate the axial component of the total angular momentum.

ii. Determine the angle between the angular momentum vector and the axle. [Hint: Compute the angular momentum vectors for both masses from a consistent reference point, possibly the center of mass.]

A thin metal plate having a mass of 0.25 g is experiencing oscillations. Determine the maximum speed attained by the plate if the frequency of oscillations is 1.5 MHz. The metal plate shows a maximum acceleration of 1.40 × 10⁸ m/s².

A simple harmonic oscillator is made up of a 0.25 kg object attached to a spring with a force constant of 100 N/m. The object's maximum position, measured from the equilibrium position, is 5 cm. Calculate the object's speed when its position is -2 cm.