between points in a plane do not change when a coordinate system is rotated In other words, the magnitude of a vector Is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle φ to become a new coordinate system S’ , as shown in the following figure. A point in a plane has coordinates ( x , y ) in S and coordinates ( x ' , y ' ) in S’. (a) Show that, during the transformation of rotation, the coordinates in S are expressed in terms of the coordinates in S by the following relations: { x ' = x cos φ + y sin φ y ' = − x sin φ + y cos φ (b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that x 2 + y 2 = x ' 2 + y ' 2 . (c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that ( x P = x Q ) 2 + ( y P − y Q ) 2 = ( x ' P = x ' Q ) 2 + ( y ' P − y ' Q ) 2 .
between points in a plane do not change when a coordinate system is rotated In other words, the magnitude of a vector Is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle φ to become a new coordinate system S’ , as shown in the following figure. A point in a plane has coordinates ( x , y ) in S and coordinates ( x ' , y ' ) in S’. (a) Show that, during the transformation of rotation, the coordinates in S are expressed in terms of the coordinates in S by the following relations: { x ' = x cos φ + y sin φ y ' = − x sin φ + y cos φ (b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that x 2 + y 2 = x ' 2 + y ' 2 . (c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that ( x P = x Q ) 2 + ( y P − y Q ) 2 = ( x ' P = x ' Q ) 2 + ( y ' P − y ' Q ) 2 .
between points in a plane do not change when a coordinate system is rotated In other words, the magnitude of a vector Is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle
φ
to become a new coordinate system S’, as shown in the following figure. A point in a plane has coordinates
(
x
,
y
)
in S and coordinates
(
x
'
,
y
'
)
in S’.
(a) Show that, during the transformation of rotation, the coordinates in S are expressed in terms of the coordinates in S by the following relations:
{
x
'
=
x
cos
φ
+
y
sin
φ
y
'
=
−
x
sin
φ
+
y
cos
φ
(b) Show that the distance of point
P
to the origin is invariant under rotations of the coordinate system. Here, you have to show that
x
2
+
y
2
=
x
'
2
+
y
'
2
.
(c) Show that the distance between points
P
and
Q
is invariant under rotations of the coordinate system. Here, you have to show that
(
x
P
=
x
Q
)
2
+
(
y
P
−
y
Q
)
2
=
(
x
'
P
=
x
'
Q
)
2
+
(
y
'
P
−
y
'
Q
)
2
.
For each of the actions depicted below, a magnet and/or metal loop moves with velocity v→ (v→ is constant and has the same magnitude in all parts). Determine whether a current is induced in the metal loop. If so, indicate the direction of the current in the loop, either clockwise or counterclockwise when seen from the right of the loop. The axis of the magnet is lined up with the center of the loop. For the action depicted in (Figure 5), indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop). I know that the current is clockwise, I just dont understand why. Please fully explain why it's clockwise, Thank you
A planar double pendulum consists of two point masses \[m_1 = 1.00~\mathrm{kg}, \qquad m_2 = 1.00~\mathrm{kg}\]connected by massless, rigid rods of lengths \[L_1 = 1.00~\mathrm{m}, \qquad L_2 = 1.20~\mathrm{m}.\]The upper rod is hinged to a fixed pivot; gravity acts vertically downward with\[g = 9.81~\mathrm{m\,s^{-2}}.\]Define the generalized coordinates \(\theta_1,\theta_2\) as the angles each rod makes with thedownward vertical (positive anticlockwise, measured in radians unless stated otherwise).At \(t=0\) the system is released from rest with \[\theta_1(0)=120^{\circ}, \qquad\theta_2(0)=-10^{\circ}, \qquad\dot{\theta}_1(0)=\dot{\theta}_2(0)=0 .\]Using the exact nonlinear equations of motion (no small-angle or planar-pendulumapproximations) and assuming the rods never stretch or slip, determine the angle\(\theta_2\) at the instant\[t = 10.0~\mathrm{s}.\]Give the result in degrees, in the interval \((-180^{\circ},180^{\circ}]\).
What are the expected readings of the ammeter and voltmeter for the circuit in the figure below? (R = 5.60 Ω, ΔV = 6.30 V)
ammeter
I =
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