An endless thin wire of density Y1 (unit: kg/m) is just above the x axis. An infinitely thin layer of density Y2 (unit: kg/m²) is parallel to the x-y plane and intersects the z-axis at the point z= -a. Find the gravitational field at (x,y,z) position. Give your answer in terms of (G,Y1,Y2, a, x, y, z, ^x, ^y, ^z(unit vectors)).

One model for a certain planet has a core of radius R and mass M surrounded by an outer shell of inner radius R, outer radius 2R, and mass 4M. If M = 4.07 × 1024 kg and R = 5.79 x 106 m, what is the gravitational acceleration of a particle at points (a) R and (b) 3R from the center of the planet? (a) Number i Units (b) Number i Units

Tidal forces are gravitational forces exerted on different parts of a object by a second object. Their effects are particularly visible on Earth's surface in the form of tides. To understand the origin of tidal forces, consider Earth-Moon system to consist of two spherical bodies, each with a spherical mass distribution. Let RE be the radius of Earth, m be the mass of the Moon, and G be the gravitational constant. Part B Since the gravitational force between two bodies decreses with distance, the accelaeration a(near) experienced by a unit mass located at the point on the earth's surface closest to moon is slightly different from the acceleration a(far) experienced by a unit mass located at the point on the earth`s surface farthest from the moon. Give a general expresion for the quantity a(near)- a(far).

Compute the gravitational attraction on a unit mass at the origin due to the mass (of constant density) occupying the volume inside the sphere r = 2a and above the plane z = a. Hint: The magnitude of the gravitational force on the unit mass due to the element of mass dM at (r, θ, φ) is (G/r2)dM. You want the z component of this since the other components of the total force are zero by symmetry. Use spherical coordinates.

Consider a thin disc of radius R and consisting of a material with constant mass density (per unit of area) g. Use cylindrical coordinates, with the z-axis perpendicular to the plane of the disc, and the origin at the disc's centre. We are going to calculate the gravitational potential, and the gravitational field, in points on the z-axis only. the gravitational potential p(2) set up by that disc is given by dr'; ()² + z² sp(2) = 27Gg

A uranium ion and an iron ion are separated by a distance of ?=57.10 nm, as shown in the figure. The uranium atom is singly ionized; the iron atom is doubly ionized.   Calculate the distance ? from the uranium atom at which an electron will be in equilibrium. Ignore the gravitational attraction between the particles. ?=   What is the magnitude ?U of the force on the electron from the uranium ion?

A skier starts from rest and slides down a slope of length L = 1080 m and angle a = 12° relative to the ground which is latitude line of λ = 57° relative to the equator. W X 2 N X L Z α Find the deflection (in meters, including sign) of the skier when it reaches the bottom of the track due to Coriolis force. Note: 1. Assume that the gravitational force is directed into the center of the Earth and it includes the centrifugal force. 2. Think about the trajectory of the skier without the effect of Coriolis force, and from there find the effect of Coriolis on the acceleration. The acceleration is time dependent and from there you can find the deflection. 3. The deflection is very small, so be very accurate with your calculation. Use g = 9.8 m/s².

Tidal forces are gravitational forces exerted on different parts of a body by a second body. Their effects are particularly visible on the earth's surface in the form of tides. To understand the origin of tidal forces, consider the earth-moon system to consist of two spherical bodies, each with a spherical mass distribution. Let r. be the radius of the earth, m be the mass of the moon, and G be the gravitational constant. Part B Since the gravitational force between two bodies decreases with distance, the acceleration anear experienced by a unit mass located at the point on the earth's surface closest to the moon is slightly different from the acceleration afar experienced by a unit mass located at the point on the earth's surface farthest from the moon. Give a general expression for the quantity anear-afar Express your answer in terms of G, m, r, and re. ▸ View Available Hint(s) anear afar = ΜΕ ΑΣΦ ? x Xb √√x √x X [X] X.10" Variables are case sensitive. No credit lost. Try again.

Field g on Different Planets 1 3. m m m,r m,2r 2 > 受,2r Four planets have relative masses and radii as shown above. The gravitational field g is smallest on the surface of which planet? 1 2 4 4. 2.

What is the smallest value of B that can be set up at the equator to permit a proton of speed 10 m/s to circulate around the earth? (R = 6.4× 106 m, m = 1.67 × 10² kg).