A horizontal block-spring system with the block on a frictionless surface has total mechanical energy E = 56.0 J and a maximum displacement from equilibrium of 0.258 m.

(a) What is the spring constant?
N/m

(b) What is the kinetic energy of the system at the equilibrium point?
J

(c) If the maximum speed of the block is 3.45 m/s, what is its mass?
kg

(d) What is the speed of the block when its displacement is 0.160 m?
m/s

(e) Find the kinetic energy of the block at x = 0.160 m.
J

(f) Find the potential energy stored in the spring when x = 0.160 m.
J

(g) Suppose the same system is released from rest at x = 0.258 m on a rough surface so that it loses 15.2 J by the time it reaches its first turning point (after passing equilibrium at x = 0). What is its position at that instant?
m

A cue ball traveling at 5.0 m/s makes a glancing, elastic collision with a target ball of equal mass that is initially at rest. The cue ball is deflected so that it makes an angle of 30° with its original direction of travel.

(a) Find the angle between the velocity vectors of the two balls after the collision.
(b) Find the speed of each ball after the collision.

One billiard ball is shot east at 1.8 m/s . A second, identical billiard ball is shot west at 1.1 m/s . The balls have a glancing collision, not a head-on collision, deflecting the second ball by 90∘ and sending it north at 1.49 m/s .

a) What is the speed of the first ball after the collision? Express your answer to two significant figures and include the appropriate units.

b) What is the direction of the first ball after the collision? Give the direction as an angle south of east. Express your answer to two significant figures and include the appropriate units.

The CM of an empty 1000-kg car is 2.45 m behind the front of the car. How far from the front of the car will the CM be when two people sit in the front seat 2.70 m from the front of the car, and three people sit in the back seat 3.80 m from the front? Assume that each person has a mass of 70.5 kg .

A bullet of mass 1.7×10−3 kg embeds itself in a wooden block with mass 0.980 kg , which then compresses a spring (k = 160 N/m ) by a distance 6.0×10−2 m before coming to rest. The coefficient of kinetic friction between the block and table is 0.53. What is the initial speed of the bullet?

Can you give me an example of translational equilibrium without rotational equilibrium?
I also need an example of rotational equilibrium without translational equilibrium. Thanks!

A 60.0-g object connected to a spring with a force constant of 40.0 N/m oscillates with an amplitude of 7.00 cm on a frictionless, horizontal surface.

(a) Find the total energy of the system.
mJ

(b) Find the speed of the object when its position is 1.30 cm. (Let 0 cm be the position of equilibrium.)
m/s

(c) Find the kinetic energy when its position is 2.50 cm.
mJ

(d) Find the potential energy when its position is 2.50 cm.
mJ

Two wheels have the same mass and radius of 4.9 kg and 0.41 m, respectively. One has (a) the shape of a hoop and the other (b) the shape of a solid disk. The wheels start from rest and have a constant angular acceleration with respect to a rotational axis that is perpendicular to the plane of the wheel at its center. Each turns through an angle of 13 rad in 9.5 s. Find the net external torque that acts on each wheel.

A torsion pendulum is made from a disk of mass m = 6.6 kg and radius R = 0.66 m. A force of F = 44.8 N exerted on the edge of the disk rotates the disk 1/4 of a revolution from equilibrium

1)What is the torsion constant of this pendulum?

2)What is the minimum torque needed to rotate the pendulum a full revolution from equilibrium?

3)What is the angular frequency of oscillation of this torsion pendulum?

4)Which of the following would change the period of oscillation of this torsion pendulum?

A) increasing the mass

B) decreasing the initial angular displacement

C) replacing the disk with a sphere of equal mass and radius

D)hanging the pendulum in an elevator accelerating downward

At 20 ° C a mineral solid in the air weighed 482.5 g and 453.8 g when immersed in water.

1)Calculate the absolute density and relative density in water.

2)If the density of benzene, at that temperature is 0.879 g / ml. Calculate the weight of the mineral when immersed in benzene

1) A string trimmer is a tool for cutting grass and weeds; it utilizes a length of nylon "string" that rotates about an axis perpendicular to one end of the string. The string rotates at an angular speed of 47 rev/s, and its tip has a tangential speed of 56 m/s. What is the length of the rotating string?

2) A planet orbits a star, in a year of length 4.62 x 10⁷ s, in a nearly circular orbit of radius 3.87 x 10¹¹ m. With respect to the star, determine (a) the angular speed of the planet, (b) the tangential speed of the planet, and (c) the magnitude of the planet's centripetal acceleration.

3) A star has a mass of 1.56 x 10³⁰ kg and is moving in a circular orbit about the center of its galaxy. The radius of the orbit is 3.9 x 10⁴ light-years (1 light-year = 9.5 x 10¹⁵ m), and the angular speed of the star is 2.3 x 10^-15 rad/s. (a) Determine the tangential speed of the star. (b) What is the magnitude of the net force that acts on the star to keep it moving around the center of the galaxy?

Over a spatial continuum, it is easy to see why some topological solitons like vortices and monopoles have to be stable. For similar reasons, Skyrmions also have to be stable, with a conserved topological density. The reason is nontrivial homotopy.

Surprisingly, in some phases, but not all phases, the analog of topological solitons, or at least what can be interpreted as them, also emerge over lattice models. Why is that? There is no nontrivial homotopy over a lattice. Why are there some phases of the XY-model with deconfined vortices and antivortices? Why are deconfined monopoles present in some 3D lattice models?

A small solid sphere of mass M₀, of radius R₀, and of uniform density ρ₀ is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that sphere is replaced by a new solid sphere of uniform density.
Read it to me
R F U R or U F or U R or F or U  The new sphere has density ρ = ρ₀ and radius R > R₀
R F U R or U F or U R or F or U  The new sphere has radius R = R₀ and density ρ > ρ₀
R F U R or U F or U R or F or U  The new sphere has density ρ < ρ₀ and mass M = M₀
R F U R or U F or U R or F or U  The new sphere has radius R > R₀ and density ρ < ρ₀
R F U R or U F or U R or F or U  The new sphere has mass M = M₀ and radius R < R₀
R F U R or U F or U R or F or U  The new sphere has mass M < M₀ and density ρ = ρ₀