Recall the definition of the cross product with respect to the unit vectors:
i × i = j × j = k × k = 0
i × j = k
j × k = i
k × i = j
and that the product is anticommutative, so that for any two vectors u and v, we have u × v = - (v × u). (This essentially takes care of part (b).)
Now, given a = 8i + j - 2k and b = 5i - 3j + k, we have
a × b = (8i + j - 2k) × (5i - 3j + k)
a × b = 40 (i × i) + 5 (j × i) - 10 (k × i)
… … … … - 24 (i × j) - 3 (j × j) + 6 (k × j)
… … … … + 8 (i × k) + (j × k) - 2 (k × k)
a × b = - 5 (i × j) - 10 (k × i) - 24 (i × j) - 6 (j × k) - 8 (k × i) + (j × k)
a × b = - 5k - 10j - 24k - 6i - 8j + i
a × b = -5i - 18j - 29k
Answer:
First law: kinetic energy is used to turn an electric generator
Second law: some thermal energy is lost to the environment as it travels through the system
Explanation:
The first law of thermodynamics is known as the law of conservation of energy. It states that energy can neither be created nor destroyed but can only be transferred or changed from one form to another. When thermal energy is used to generate electricity, the kinetic energy of the steam is used to turn the electric generator (thereby producing electrical energy).
The second law of thermodynamics states that energy transfer or transformation leads to an increase in entropy resulting in the loss of energy. This law also states that as energy is transferred or transformed, some is lost in a form that is unusable. When thermal energy is used to generate electricity, some of the thermal energy is lost to the environment as it travels through the system.
Answer:
a. 0.2 A
Explanation:
Given;
voltage of the battery, V = 12 V
three resistance connected in series, R₁ = 15 Ω, R₂ = 21 Ω, R₃ = 24Ω
The equivalent resistance for series connection is calculated as;
Rt = R₁ + R₂ + R₃
Rt = 15Ω + 21Ω + 24Ω
Rt = 60 Ω
Apply Ohm's law to calculate the current n the circuit;
V = IRt
I = V/Rt
I = 12/60
I = 0.2 A
Therefore, the current in the circuit is 0.2 A
Answer:
The solid sphere will reach the bottom first.
Explanation:
In order to develop this problem and give it a correct solution, it is necessary to collect the concepts related to energy conservation. To apply this concept, we first highlight the importance of conserving energy so we will match the final and initial energies. Once this value has been obtained, we will concentrate on finding the speed, and solving what is related to the Inertia.
In this way we know that,
We know as well that the lineal and angular energy are given by,
And the tangential kinetic energy as
Where
Replacing
Re-arrange for v,
We have here three different objects: solid cylinder, hollow pipe and solid sphere. We need the moment inertia of this objects and replace in the previous equation found, then,
For hollow pipe:
For solid cylinder:
For solid sphere,
Then comparing the speed of the three objects we have:
Answer:
0.25 m/s
Explanation:
From the law of conservation of momentum
Mu+ mu = Mv' + m v
M= mass of the astronaut = 80 kg
m= mass of the oxygen tank= 10 Kg
v= speedof the tank 2 m/s
u= initial velocity of the system= 0
If we substitute the values, we have
( 80× 0 )+(10×0)= [(80 x v )+ (10 x 2)]
0= 80v + 20
-20=80v
v= -0.25 m/s ( we have a negative value because the astronaut and the motion of the cylinder are in opposite direction)
Hence the velocity the astronaut start to move off into space is 0.25 m/s
