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physics
modern physics

Questions and Answers of Modern Physics

Show that the number density of dust measured by an arbitrary observer whose four-velocity is is -.
Complete the proof that Eq. (4.14) defines a tensor by arguing that it must be linear in both its arguments.
Establish Eq. (4.19) from the preceding equations.In Eq.4.19Dust:
Show that Eq. (4.34), when α is any spatial index, is just Newton's second law. In Eq. (4.34) T((,( = 0.
Show that Eq. (4.34), when α is any spatial index, is just Newton's second law. In Eq. (4.34) T((,( = 0. Discuss.
Prove that, defined in Eq. (5.52), is a tensor.
For those who have done both Exers. 11 and 12, show in polars that
For the tensor whose polar components are (Arr = r2, Ar( = r sin (, A(r = r cos (, A(( = tan (), compute Eq. (5.65) in polars for all possible indices.
For the vector whose polar components are (Vr = 1, VÎ¸ = 0), compute in polars all components of the second covariant derivative VÎ±;Î¼;Î½. To find
Discover how each expression V(,( and V((((( separately transforms under a change of coordinates (for (((, begin with Eq. (544)). Show that neither is the standard tensor law, but that their sum does
Show that if U( (( V( = W( then U( (( V( = W(.
(a) Show that the coordinate transformation (x, y) ( ξ with ξ = x and ( = 1. ((/(x = 0 and (( / (y = 0. This violates Eq. (5.6). (b) Are the following coordinate transformations good ones? Compute
A curve is defined by {x = ( (λ), y = g(λ), 0 ( ( ( 1}. Show that the tangent vector (dx/dλ, dy/dλ) does actually lie tangent to the curve.
Justify the pictures in Fig. 5.5.
Calculate all elements of the transformation matrices and for the transformation from Cartesian (x, y) - the unprimed indices - to polar (r, Î¸) - the primed indices.
Draw a diagram similar to Fig. 5.6 to explain Eq. (5.38).
Decide if the following sets are manifolds and say why. If there are exceptional points at which the sets are not manifolds, give them: (a) Phase space of Hamiltonian mechanics, the space of the
A 'straight line' on a sphere is a great circle, and it is well known that the sum of the interior angles of any triangle on a sphere whose sides are arcs of great circles exceeds 180—¦.
In this exercise we will determine the condition that a vector field can be considered to be globally parallel on a manifold. More precisely, what guarantees that we can find a vector field
Prove that Eq. (6.52) defines a new affine parameter. ϕ = aλ + b,
The proper distance along a curve whose tangent is is given by Eq. (6.8). Show that if the curve is a geodesic, then proper length is an affine parameter. (Use the result of Exer. 13.)
(a) Derive Eqs. (6.59) and (6.60) from Eq. (6.58).(b) Fill in the algebra needed to justify Eq. (6.61).Eq. (58)Eq. (59) Eq. (61) Î´VÎ± = Î´a Î´b
Prove that RÎ±Î²Î¼Î½ = 0 for polar coordinates in the Euclidean plane. Use Eq. (5.45) or equivalent results.Eq. (5.45)
Of the manifolds in Exer. 1, on which is it customary to use a metric, and what is that metric? On which would a metric not normally be defined, and why? The interior of a circle of unit radius in
Fill in the algebra necessary to establish Eq. (6.73).
Consider the sentences following Eq. (6.78). Why does the argument in parentheses not apply to the signs in Vα;β = Vα,β + ΓαμβVμ and Vα;β = Vα;β = Vα;β = Vα,β - ΓμαβVμ?
Prove Eq. (6.88). (Be careful: one cannot simply differentiate Eq. (6.67) since it is valid only at P, not in the neighborhood of P.)Eq. (6.67)RÎ±Î²Î¼v = 1/2
Establish Eq. (6.89) from Eq. (6.88). Eq. (6.88) Rαβμv,λ = 1/2 (gαv,βμλ - gαμ,βμλ + gβμ,αvλ - gβv,αμλ). Eq. (6.89) Rαβμv,λ + Rαβλμ,v + Rαβvλ,μ = 0
(a) Prove that the Ricci tensor is the only independent contraction of Rαβμν: all others are multiples of it. (b) Show that the Ricci tensor is symmetric.
In polar coordinates, calculate the Riemann curvature tensor of the sphere of unit radius, whose metric is given in Exer. 28. (Note that in two dimensions there is only one independent component, by
Show that covariant differentiation obeys the usual product rule, e.g. (VαβWβγ);μ = Vαβ ;μ Wβγ + VαβWβγ ;μ.
A four-dimensional manifold has coordinates (u, v, w, p) in which the metric has components guv = gww = gpp = 1, all other independent components vanishing. (a) Show that the manifold is flat and the
A 'three-sphere' is the three-dimensional surface in four-dimensional Euclidean space (coordinates x, y, z, w), given by the equation x2 + y2 + z2 + w2 = r2, where r is the radius of the sphere. (a)
Prove the following results used in the proof of the local flatness theorem in § 6.2: (a) The number of independent values of ∂2xα/∂xγ'∂xμ'|0 is 40. (b) The corresponding number for
(a) Prove that Î“Î¼Î±Î² = Î“Î¼Î²Î± in any coordinate system in a curved Riemannian space.(b) Use this
Prove that the first term in Eq. (6.37) vanishes.Eq.(6.37)
Fill in the missing algebra leading to Eqs. (6.40) and (6.42).Eq. (40)Î“Î± Î¼Î± = (œ“ - gVÎ±),Î±).Eq. (42)
Show that Eq. (6.42) leads to Eq. (5.56).Eq. (42)Eq. (56)
If Eq. (7.3) were the correct generalization of Eq. (7.1) to a curved spacetime, how would you interpret it? What would happen to the number of particles in a commoving volume of the fluid, as time
To first order in ϕ, compute gαβ for Eq. (7.8). ds2 = − (1 + 2ϕ) dt2 + (1 - 2ϕ) (dx2 + dy2 + dz2).
Verify that the results, Eqs. (7.15) and (7.24), depended only on g00: the form of gxx doesn't affect them, as long as it is 1 + 0(ϕ). Eq. (7.15) d/dτ p0 = - m ∂ϕ/ ∂τ. Eq. (7.24) dpi / dτ =
Deduce Eq. (7.25) from Eq. (7.10). Eq. (7.10) ∇β = 0. Eq. (7.25) pαpβ;α = 0
Show that Eq. (8.2) is a solution of Eq. (8.1) by the following method. Assume the point particle to be at the origin, r = 0, and to produce a spherically symmetric field. Then use Gauss' law on a
(a) Derive the following useful conversion factors from the SI values of G and c:G/c2 = 7.425 Ã— 10ˆ’28mkgˆ’1 = 1,c5/G = 3.629 Ã— 1052J sˆ’1 =
(a) Calculate in geometrized units: (i) The Newtonian potential φ of the Sun at the Sun's surface, radius 6.960 × 108m; (ii) The Newtonian potential φ of the Sun at the radius of Earth's orbit, r
(a) Let A be an n Ã— n matrix whose entries are all very small, |Aij| 1/n, and let I be the unit matrix. Show that(I + A)ˆ’1 = I ˆ’ A + A2 ˆ’ A3 + A4
The wheels on a moving bicycle have both translational (or linear) and rotational motions. What is meant by the phrase "a rigid body, such as a bicycle wheel, is in equilibrium"? (a) The body cannot
The drawing shows three objects rotating about a vertical axis. The mass of each object is given in terms of m0, and its perpendicular distance from the axis is specified in terms of r0. Rank the
The same force is applied to the edge of two hoops (see the drawing). The hoops have the same mass, whereas the radius of the larger hoop is twice the radius of the smaller one. The entire mass of
Two hoops, starting from rest, roll down identical inclined planes. The work done by nonconservative forces, such as air resistance, is zero (Wnc = 0 J). Both have the same mass M, but, as the
An ice skater is spinning on frictionless ice with her arms extended outward. She then pulls her arms in toward her body, reducing her moment of inertia. Her angular momentum is conserved, so as she
The drawing illustrates an overhead view of a door and its axis of rotation. The axis is perpendicular to the page. There are four forces acting on the door, and they have the same magnitude. Rank
Five hockey pucks are sliding across frictionless ice. The drawing shows a top view of the pucks and the three forces that act on each one. As shown, the forces have different magnitudes (F, 2F, or
The drawing shows a top view of a square box lying on a frictionless floor. Three forces, which are drawn to scale, act on the box. Consider an angular acceleration with respect to an axis through
A rotational axis is directed perpendicular to the plane of a square and is located as shown in the drawing. Two forces, 1 and 2 , are applied to diagonally opposite corners, and act along the sides
A person is standing on a level floor. His head, upper torso, arms, and hands together weigh 438 N and have a center of gravity that is 1.28 m above the floor. His upper legs weigh 144 N and have a
The drawing shows a person (weight, W = 584 N) doing push-ups. Find the normal force exerted by the floor on each hand and each foot, assuming that the person holds this position.
A person exerts a horizontal force of 190 N in the test apparatus shown in the drawing. Find the horizontal force (magnitude and direction) that his flexor muscle exerts on his forearm.
The drawing shows a rectangular piece of wood. The forces applied to corners B and D have the same magnitude of 12 N and are directed parallel to the long and short sides of the rectangle. The long
The wheels, axle, and handles of a wheelbarrow weigh 60.0 N. The load chamber and its contents weigh 525 N. The drawing shows these two forces in two different wheelbarrow designs. To support the
The steering wheel of a car has a radius of 0.19 m, and the steering wheel of a truck has a radius of 0.25 m. The same force is applied in the same direction to each steering wheel. What is the ratio
See Example 4 for data pertinent to this problem. What is the minimum value for the coefficient of static friction between the ladder and the ground, so that the ladder (with the fireman on it) does
The drawing shows a uniform horizontal beam attached to a vertical wall by a frictionless hinge and supported from below at an angle Î¸ = 39o by a brace that is attached to a pin. The
A man holds a 178-N ball in his hand, with the forearm horizontal (see the drawing). He can support the ball in this position because of the flexor muscle force , which is applied perpendicular to
The drawing shows a bicycle wheel resting against a small step whose height is h = 0.120 m. The weight and radius of the wheel are W = 25.0 N and r = 0.340 m, respectively. A horizontal force is
A person is sitting with one leg outstretched and stationary, so that it makes an angle of 30.08 with the horizontal, as the drawing indicates. The weight of the leg below the knee is 44.5 N, with
A wrecking ball (weight = 4800 N) is supported by a boom, which may be assumed to be uniform and has a weight of 3600 N. As the drawing shows, a support cable runs from the top of the boom to the
A man drags a 72-kg crate across the floor at a constant velocity by pulling on a strap attached to the bottom of the crate. The crate is tilted 25° above the horizontal, and the strap is inclined
The drawing shows an A-shaped stepladder. Both sides of the ladder are equal in length. This ladder is standing on a frictionless horizontal surface, and only the crossbar (which has a negligible
Consult Multiple-Concept Example 10 to review an approach to problems such as this. A CD has a mass of 17 g and a radius of 6.0 cm. When inserted into a player, the CD starts from rest and
A clay vase on a potter's wheel experiences an angular acceleration of 8.00 rad/s2 due to the application of a 10.0-N ∙ m net torque. Find the total moment of inertia of the vase and potter's wheel.
A solid circular disk has a mass of 1.2 kg and a radius of 0.16 m. Each of three identical thin rods has a mass of 0.15 kg. The rods are attached perpendicularly to the plane of the disk at its outer
A ceiling fan is turned on and a net torque of 1.8 N ∙ m is applied to the blades. The blades have a total moment of inertia of 0.22 kg ∙ m2. What is the angular acceleration of the blades?
Multiple-Concept Example 10 provides one model for solving this type of problem. Two wheels have the same mass and radius of 4.0 kg and 0.35 m, respectively. One has the shape of a hoop and the other
A 9.75-m ladder with a mass of 23.2 kg lies flat on the ground. A painter grabs the top end of the ladder and pulls straight upward with a force of 245 N. At the instant the top of the ladder leaves
Multiple-Concept Example 10 offers useful background for problems like this. A cylinder is rotating about an axis that passes through the center of each circular end piece. The cylinder has a radius
A long, thin rod is cut into two pieces, one being twice as long as the other. To the midpoint of piece A (the longer piece), piece B is attached perpendicularly, in order to form the inverted "T"
Two children hang by their hands from the same tree branch. The branch is straight, and grows out from the tree trunk at an angle of 27.0o above the horizontal. One child, with a mass of 44.0 kg, is
Multiple-Concept Example 10 reviews the approach and some of the concepts that are pertinent to this problem. The drawing shows a model for the motion of the human forearm in throwing a dart. Because
A 15.0-m length of hose is wound around a reel, which is initially at rest. The moment of inertia of the reel is 0.44 kg ∙ m2, and its radius is 0.160 m. When the reel is turning, friction at the
The drawing shows two identical systems of objects; each consists of the same three small balls connected by massless rods. In both systems the axis is perpendicular to the page, but it is located at
The drawing shows the top view of two doors. The doors are uniform and identical. Door A rotates about an axis through its left edge, and door B rotates about an axis through its center. The same
The parallel axis theorem provides a useful way to calculate the moment of inertia I about an arbitrary axis. The theorem states that I = Icm + Mh2, where Icm is the moment of inertia of the object
The crane shown in the drawing is lifting a 180-kg crate upward with an acceleration of 1.2 m/s2. The cable from the crate passes over a solid cylindrical pulley at the top of the boom. The pulley
Calculate the kinetic energy that the earth has because of(a) Its rotation about its own axis(b) Its motion around the sun. Assume that the earth is a uniform sphere and that its path around the sun
A helicopter has two blades (see Figure 8.11); each blade has a mass of 240 kg and can be approximated as a thin rod of length 6.7 m. The blades are rotating at an angular speed of 44 rad/s.(a) What
A solid sphere is rolling on a surface. What fraction of its total kinetic energy is in the form of rotational kinetic energy about the center of mass?
Starting from rest, a basketball rolls from the top of a hill to the bottom, reaching a translational speed of 6.6 m/s. Ignore frictional losses.(a) What is the height of the hill?(b) Released from
One end of a thin rod is attached to a pivot, about which it can rotate without friction. Air resistance is absent. The rod has a length of 0.80 m and is uniform. It is hanging vertically straight
A bowling ball encounters a 0.760-m vertical rise on the way back to the ball rack, as the drawing illustrates. Ignore frictional losses and assume that the mass of the ball is distributed uniformly.
A tennis ball, starting from rest, rolls down the hill in the drawing. At the end of the hill the ball becomes airborne, leaving at an angle of 35° with respect to the ground. Treat the ball as a
A square, 0.40 m on a side, is mounted so that it can rotate about an axis that passes through the center of the square. The axis is perpendicular to the plane of the square. A force of 15 N lies in
When some stars use up their fuel, they undergo a catastrophic explosion called a supernova. This explosion blows much or all of the star's mass outward, in the form of a rapidly expanding spherical
Just after a motorcycle rides off the end of a ramp and launches into the air, its engine is turning counterclockwise at 7700 rev/min. The motorcycle rider forgets to throttle back, so the engine's
A thin rod has a length of 0.25 m and rotates in a circle on a frictionless tabletop. The axis is perpendicular to the length of the rod at one of its ends. The rod has an angular velocity of 0.32
As seen from above, a playground carousel is rotating counterclockwise about its center on frictionless bearings. A person standing still on the ground grabs onto one of the bars on the carousel very
A cylindrically shaped space station is rotating about the axis of the cylinder to create artificial gravity. The radius of the cylinder is 82.5 m. The moment of inertia of the station without people
A thin, uniform rod is hinged at its midpoint. To begin with, one-half of the rod is bent upward and is perpendicular to the other half. This bent object is rotating at an angular velocity of 9.0
A small 0.500-kg object moves on a frictionless horizontal table in a circular path of radius 1.00 m. The angular speed is 6.28 rad/s. The object is attached to a string of negligible mass that
A platform is rotating at an angular speed of 2.2 rad/s. A block is resting on this platform at a distance of 0.30 m from the axis. The coefficient of static friction between the block and the

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