1. a centripetal-acceleration addict rides in uniform circular motion with period T=2s and radius r=3m. at t1 his acceleration vector= 6(m/s^2)i + -4(m/s^2)j. At that instant, what are the values of vector b (dot) vector a
and vector r (dot) vector a
All I could figure out was that through this formula: T=(2(pi))r/T
V=9.425m/s
how do you make that into a vector though? and how do you find the r vector? is there a way to find the angle theta?
Vectors r, a and b have not been defined in the question. Perhaps these are notations used in your class and are obvious to you.
Please define what they are (velocity, acceleration, radial, tangential...)
I suppose unit vectors i, j are parallel to the x,y-axes respectively.
You have correctly calculated the tangential speed of
V=9.425m/s (=2pi*r/T)
The answer is zero because the angle between the two vectors is 90. Cos(90)= 0
To find the dot product of two vectors, you need to multiply their corresponding components and then sum the products.
Let's first express the velocity vector as a vector quantity. The magnitude of the velocity can be found using the formula you provided: V = 2πr/T. In this case, V = 2π(3 m)/(2 s) = 3π m/s.
To find the direction of the velocity vector, we need its components. We can break down the velocity vector into its x and y components using trigonometry. The x-component of the velocity vector can be found by multiplying the magnitude by the cosine of the angle. Similarly, the y-component can be found by multiplying the magnitude by the sine of the angle.
For the x-component:
Vx = V * cos(θ)
= (3π m/s) * cos(θ)
For the y-component:
Vy = V * sin(θ)
= (3π m/s) * sin(θ)
Now, to find the dot product of vector b and vector a, you multiply their corresponding components and sum them:
b · a = (6 m/s²) * (3π m/s) + (-4 m/s²) * (0)
= 18π m²/s²
To find the dot product of vector r and vector a, you need to know the angle between them, expressed in radians. Since we don't have the angle information in this specific question, it seems that we can't determine this dot product without further information.
To find the vector r, you need to know its components. The vector r represents the position of the object in its circular motion. It can be represented by r = x i + y j, where i and j are the unit vectors in the x and y directions, respectively. However, without additional information, we cannot determine the specific values of vector r or its angle theta.
In summary, given the information provided, you can find the dot product of vector b and vector a, but you cannot determine the dot product of vector r and vector a nor find the vector r or the angle theta without further information.