Figure 1 shows a Newton's cradle with small beads hanging from different heights. One bead hangs 30.0 cm below the upper bar, and the second bead hangs 20.0 cm below the lower bar, so that the beads are aligned horizontally. (In reality, the beads hang from two strings attached to them in a V-formation, so that the beads can only swing in the plane of the page; still, the beads behave like simple pendulums with the lengths shown.) The beads have equal masses, and collisions between them are always elastic. 30.0 cm 20.0 cm

After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 46.0 cm. The explorer finds that the pendulum completes 98.0 full swing cycles in a time of 145s. What is the magnitude of the gravitational acceleration on this planet? gPlanet=(?)m/s^2

A frictionless pendulum of mass m=1 kg and length L=1 m, gets released at t=0 from a distance x=0.12 m relative to its equilibrium position. м M What is the velocity of the pendulum's mass at time t=1.5 s? Give your answer in units of [m/s] and up to 1 decimal place (e.g. 0.6)

An orange block of mass 2.5 kg is on an inclined plane. There is no friction between the block and the plane except in the region shown in red and black on the diagram above.  The plane is inclined at an angle of θ = 17 degrees above horizontal. The block begins at rest, leaning against a spring with a spring constant of k = 741 newtons per meter.  At the start, the spring has a compressed length of x1 = 0.39 meters.  The spring then expands to its equilibrium length of x2 = 0.65 meters, sending the block sliding up the ramp. The block has no friction with the plane except in the single region of length L = 0.78 meters, shown in red and black above. The coefficient of kinetic friction between the block and this region is μ K = 0.24. The block slides past this region and continues up the plane. What is the speed of the block, in units of meters per second, when it has slid a distance d = 1.3 meters along the plane from the place where it started? (This is the distance measured from the…

An 12 lb ball attached to the end of a 14 ft long string is rotating as a conical pendulum. If it rotates at 40 rev/min, what angle will the string make with the vertical?

A spring with spring constant k and equilibrium length zero is attached tothe top of a frictionless hoop of radius R. The spring is stretched and connected to a bead of mass m at the bottom of the hoop. At t = 0 the bead is given an initial speed v0 and the bead moves up the hoop. Find the speed of the bead as a function of position on hoop v(θ).

The “Giant Swing” at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a cable 5.00 m long, and the upper end of the cable is fastened to the arm at a point 3.00 m from the central shaft (Fig. 3). (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of 30.0° with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution?

A simple pendulum is suspended from the ceiling by means of a string and has a length of 1.56 m long. Suppose you start with the pendulum hanging vertically, at rest. You then give it a push so that it starts swinging with a speed of 1.73 m/s. What maximum angle (in degrees) will it reach, with respect to the vertical, before falling back down?     38.3 degrees     31.9 degrees     25.5 degrees     20.4 degrees

A point-like mass m is constrained to move along the z-axis only. It is attached to two springs, each of spring constant k, and each of rest length 0. One spring has one of its ends fixed at x = 0 and y = h; the other spring is fixed at one end to z = 0 and y=-h as shown in the figure below. Gravity does not act on the system. The springs do not bend but contract and extend along straight lines between their end points. y m (d) Determine the equilibrium position of the system and the frequencies of small oscillations about this equilibrium position. Hint: use approximations (i.e. Taylor expansion) to simplify the equation of motion near the equilibrium position to that of a harmonic oscillator. (e) Repeat part (d) for the case where the two springs have different spring constants k₁ and k₂ and are placed at different distances on the y-axis h₁ and h₂.

Physics students design an experiment to measure assorted coefficients of friction using a set-up similar to the one shown (not drawn to scale). A small balloon containing sand with a combined mass of 98.2gg is suspended with an ideal string of length 86.6cmcm, forming a simple pendulum. The pendulum is released from an angle of 41.4∘∘ with the vertical, and it collides with a block of mass 333gg located where the pendulum string becomes vertical. Because the balloon weighted with sand is soft and deformable, the collision is totally inelastic. The block slides for a distance of 30.8mmmm before coming to rest. a) Calculate the speed, in meters per second, of the sand-filled balloon at the instant prior to the collision.  b) Calculate the speed, in meters per second, of the block at the instant following the collision.  c)