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physics
modern physics

Physics 10th edition David Young, Shane Stadler - Solutions

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 the second derivative, treat the first derivative VÎ±;Î¼ as any tensor:
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 obey the standard law.
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 the Jacobian and list any points at which the transformations fail. (i) ξ = (x2 + y2)1/2, η =
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 canonical coordinates and momenta pi and qi; (b) The interior of a circle of unit radius in
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—¦. Show that the amount by which a vector is rotated by parallel transport around such a triangle (as
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 satisfying the equation
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 [Î“Î±Î¼1,2 - Î“Î±Î¼2,1 +
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 two-dimensional Euclidean space;
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 gÏƒÏƒ (gÏƒv, Î²Î¼ -
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 the same arguments as in Exer. 18(b). So calculate Rθφθφ and obtain all other components in
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 signature is + 2. (b) The result in (a) implies the manifold must be Minkowski spacetime. Find a
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) Define new coordinates (r, θ, φ, χ) by the equations w = r cos χ, z = r sin χ cos θ, x = r sin
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 ∂3xα/∂xλ' ∂xμ' ∂xv'|0 is 80. (c) The corresponding number for gαβ,γ'μ'|0 is 100.
(a) Prove that Î“Î¼Î±Î² = Î“Î¼Î²Î± in any coordinate system in a curved Riemannian space.(b) Use this to prove that Eq. (6.32) can be derived in the same manner as in flat space.Eq.(6.32)
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 evolves? In principle, can we distinguish experimentally between Eqs. (7.2) and (7.3)? Eq. (7.2) (n
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 = - mϕ, jδij,
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 sphere of radius r to conclude dϕ/dr = Gm/r2. Deduce Eq. (8.2) from this. (Consider the behavior at
(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 = 1.(b) Derive the values in geometrized units of the constants in Table 8.1 from their given values in
(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 = 1AU = 1.496 × 1011m; (iii) The Newtonian potential φ of Earth at its surface, radius = 6.371 ×
(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 ˆ’ +. . .By proving that (i) the series on the right-hand side converges absolutely for each of
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 have translational or rotational motion of any kind. (b) The body can have translational motion,
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 three objects according to their moments of inertia, largest to smallest.(a) A, B, C(b) A, C, B(c) B,
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 each hoop is concentrated at its rim, so the moment of inertia is I = Mr2, where M is the mass and r
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 drawing shows, one hoop has twice the radius of the other. The moment of inertia for each hoop is I = Mr2,
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 reduces her moment of inertia, her angular velocity increases and she spins faster. Compared to her
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 3F), and are applied at different points on the pucks. Only one of the five pucks can be in
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 the center of the box (perpendicular to the page). Which one of the following statements is
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 of the square, first as shown in part a and then as shown in part b of the drawing. In each case the
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 center of gravity that is 0.760 m above the floor. Finally, his lower legs and feet together weigh 87
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 side of the rectangle is twice as long as the short side. An axis of rotation is shown perpendicular
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 wheelbarrow in equilibrium, the man's hands apply a force to the handles that is directed vertically
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 of the torque produced by this force in the truck to the torque produced in the car?
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 not slip?
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 beam has a weight of 340 N. Three additional forces keep the beam in equilibrium. The brace applies a
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 forearm. The forearm weighs 22.0 N and has a center of gravity as indicated. Find The
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 applied to the axle of the wheel. As the magnitude of increases, there comes a time when the wheel
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 the center of gravity located below the knee joint. The leg is being held in this position because of
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 tractor. The angle between the support cable and the horizontal is 328, and the angle between the boom
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 61Â° above the horizontal. The center of gravity of the crate coincides with its geometrical
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 mass) of the "A" keeps the ladder from collapsing. The ladder is uniform and has a mass of 20.0 kg.
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 accelerates to an angular velocity of 21 rad/s in 0.80 s. Assuming the CD is a uniform solid disk, determine
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 edge to form a three-legged stool (see the drawing). Find the moment of inertia of the stool with
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 the shape of a solid disk. The wheels start from rest and have a constant angular acceleration with
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 the ground, the ladder experiences an angular acceleration of 1.80 rad/s2 about an axis passing
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 of 0.0830 m, an angular speed of 76.0 rad/s, and a moment of inertia of 0.615 kg ∙ m2. A brake
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" shown in the drawing. The application of a net external torque causes this object to rotate about axis
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 hanging 1.30 m along the branch from the tree trunk. The other child, with a mass of 35.0 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 of the force applied by the triceps muscle, the forearm can rotate about an axis at the elbow
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 axle exerts a torque of magnitude 3.40 N ∙ m on the reel. If the hose is pulled so that the tension
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 a different place, as shown. The same force of magnitude F is applied to the same ball in each
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 force is applied perpendicular to each door at its right edge, and the force remains perpendicular as
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 relative to an axis that passes through the center of mass and is parallel to the axis of interest, M
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 has a mass of 130 kg. The cable is then wound onto a hollow cylindrical drum that is mounted on the
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 is circular. For comparison, the total energy used in the United States in one year is about 1.1 ×
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 is the total moment of inertia of the two blades about the axis of rotation?(b) Determine the
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 rest at the same height, a can of frozen juice rolls to the bottom of the same hill. What is the
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 downward. The end of the rod nearest the floor is given a linear speed v0, so that the rod begins to
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. The translational speed of the ball is 3.50 m/s at the bottom of the rise. Find the translational
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 thin-walled spherical shell, and determine the range x.
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 this plane and is applied to the square. What is the magnitude of the maximum torque that such a
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 shell. As a simple model of the supernova process, assume that the star is a solid sphere of radius
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 angular speed increases to 12 500 rev/min. As a result, the rest of the motorcycle (including the
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 rad/s and a moment of inertia of 1.1 × 10-3 kg ∙ m2. A bug standing on the axis decides to crawl
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 close to its outer edge and climbs aboard. Thus, this person begins with an angular speed of zero
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 is 3.00 × 109 kg ∙ m2. Suppose that 500 people, with an average mass of 70.0 kg each, live on
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 rad/s about an axis that is perpendicular to the left end of the rod and parallel to the rod's upward
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 passes through a small hole in the table at the center of the circle. Someone under the table begins to
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 platform is 0.75. Without any external torque acting on the system, the block is moved toward the axis.

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