By standard convention, both the electric potential and the the electric potential energy between two charges is taken to be zer
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To solve this problem we will apply the concepts related to wave velocity as a function of the tension and linear mass density. This is
Here
v = Wave speed
T = Tension
= Linear mass density
From this proportion we can realize that the speed of the wave is directly proportional to the square of the tension
Therefore, if there is an increase in tension of 4, the velocity will increase the square root of that proportion
The factor that the wave speed change is 2.
a) Vector product: |d1 × d2| = (34.2 m) k
b) Scalar product: d1 · d2 = -0.874 m
c) (d1 + d2) · d2 = 16.1 m
d) Component of d1 along direction of d2: -0.21 m
Explanation:
a)
In this part, we want to calculate
|d1 × d2|
Which is the vector product between the two displacements d1 and d2.
The two vectors are:
d1 = (6.00 m)i + (5.74 m)j
d2 = (-2.92 m)i + (2.9 m)j
The vector product of two vectors and
is also a vector which has components:
We notice immediatly that in this problem ,the two vectors d1 and d2 lie in the x-y plane, so they do not have components in zero. Therefore, the vector product has only one component, which is the one in z, and it is:
Therefore, the vector product of d1 and d2 is:
|d1 × d2| = (34.2 m) k
b)
In this case, we want to calculate
d1 · d2
Which is the scalar product between the two displacements.
The scalar product of two vectors and
is a scalar given by:
In this problem,
d1 = (6.00 m)i + (5.74 m)j
d2 = (-2.92 m)i + (2.9 m)j
Therefore, the scalar product between the two vectors is:
c)
In this case, we want to calculate
(d1 + d2) · d2
Which means that first we have to calculate the resultant displacement d1 + d2, and then calculate the scalar product of the resultant vector with d2.
Given two vectors and
, the resultant vector is also a vector given by
In this case,
d1 = (6.00 m)i + (5.74 m)j
d2 = (-2.92 m)i + (2.9 m)j
So the resultant vector is
So
And calculating the scalar product with d2, we find:
d)
The component of a vector a along another vector b is given by
where is the scalar product between and b
is the magnitude of vector b
In this problem, we have the two vectors
d1 = (6.00 m)i + (5.74 m)j
d2 = (-2.92 m)i + (2.9 m)j
We want to find the component of d1 along the direction of d2.
We already calculated the scalar product of the two vectors in part b):
d1 · d2 = -0.874 m
The magnitude of a vector b is given by
So, for vector d2,
Now we can calculate the component of d1 along d2:
Learn more about operations with vectors:
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