Two forces, F → 1 = ( − 6.00 i ^ − 4.00 j ^ ) N and F → 2 = ( − 3.00 i ^ + 7.00 j ^ ) N , act on a particle of mass 2.00 kg that is initially at rest at coordinates (−2.00 m, + 4.00 m). (a) What are the components of the particle’s velocity at t = 10.0 s? (b) In what direction is the particle moving at t = 10.0 s? (c) What displacement does the particle undergo during the first 10.0 s? (d) What are the coordinates of the particle at t = 10.0 s?
Two forces, F → 1 = ( − 6.00 i ^ − 4.00 j ^ ) N and F → 2 = ( − 3.00 i ^ + 7.00 j ^ ) N , act on a particle of mass 2.00 kg that is initially at rest at coordinates (−2.00 m, + 4.00 m). (a) What are the components of the particle’s velocity at t = 10.0 s? (b) In what direction is the particle moving at t = 10.0 s? (c) What displacement does the particle undergo during the first 10.0 s? (d) What are the coordinates of the particle at t = 10.0 s?
Solution Summary: The author explains the particle's velocity at t=10.0mathrmsec and the two forces acting on it.
Two forces,
F
→
1
=
(
−
6.00
i
^
−
4.00
j
^
)
N
and
F
→
2
=
(
−
3.00
i
^
+
7.00
j
^
)
N
, act on a particle of mass 2.00 kg that is initially at rest at coordinates (−2.00 m, + 4.00 m). (a) What are the components of the particle’s velocity at t = 10.0 s? (b) In what direction is the particle moving at t = 10.0 s? (c) What displacement does the particle undergo during the first 10.0 s? (d) What are the coordinates of the particle at t = 10.0 s?
(a)
Expert Solution
To determine
The components of the velocity of the particle at t=10.0sec.
Answer to Problem 5.15P
The x components of the particle’s velocity at t=10.0sec is −45.0m/s and the y component of particle’s velocity at t=10s is 15.0m/s.
Explanation of Solution
The mass of the particle is 2.00kg and the two forces acting on it are F→1=(−6.00i^−4.00j^)N and F→2=(−3.00i^+7.00j^)N. The particle is initially at rest at (−2.00m, +4.00m).
Write the formula to calculate net force acts on a particle
F→net=F→1+F→2
Here, F→net is the net force acting on a particle, F→1 and F→2 are the two given forces.
Write the formula to calculate acceleration of a particle
a→=F→netm
Here, a→ is the acceleration of the particle and m is the mass of the particle.
Substitute F→1+F→2 for F→net in above equation.
a→=F→1+F→2m
Substitute (−6.00i^−4.00j^)N for F→1, (−3.00i^+7.00j^)N for F→2 and 2.00kg for m to find a→.
Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
State whether this universe is spatially open, closed or flat.
[2 marks]
b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function
of time t, starting at t = 0.
[3 marks]
c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy
B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect
to galaxy A.
d) The Friedmann equations are
2
k
8πG
а
4πG
+
a²
(p+3p).
3
a
3
[5 marks]
Use these equations to determine the energy density p(t) and the pressure p(t) for the
FLRW spacetime specified at the top of the page.
[5 marks]
e) Given the result of question B3.d, state whether the FLRW universe in question is (i)
radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv)
none of the previous. Justify your answer.
f)
[5 marks]
A conformally…
SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
Page 2
SECTION A
Answer ALL questions in Section A
[Expect to use one single-sided A4 page for each Section-A sub question.]
Question A1
SPA6308 (2024)
Consider Minkowski spacetime in Cartesian coordinates th
=
(t, x, y, z), such that
ds² = dt² + dx² + dy² + dz².
(a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V.
(b) Consider now the coordinate system x' (u, v, y, z) such that
u =t-x,
v=t+x.
[2 marks]
Write down the line element, the metric, the Christoffel symbols and the Riemann curvature
tensor in the new coordinates. [See the Appendix of this document.]
[5 marks]
(c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify
explicitly that V. V is invariant under the coordinate transformation.
Question A2
[5 marks]
Suppose that A, is a covector field, and consider the object
Fv=AAμ.
(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
coordinate transformation.
[5 marks]
(b)…
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