A hockey puck of diameter 3 inches is spinning around its center at a speed of 3 counterclockwise rotations per second. The center of the puck is traveling at a speed of 24 inches per second at an angle of 45 degrees to the positive x-axis.
(a) At time t=0, the center of the puck is at the origin. Where is the center of the puck at time t? (Time t is measured in seconds.)
(b) A point on the outer edge of the puck begins at the point (0,3/2). What is its location at time t?
for the center:
x = 24 cos 45 * t = 17 t
y = 24 sin 45 * t = 17 t
relative to center
initial angle of point from x axis
Ai = 90 degrees (on y axis)
r = 1.5
Xrel = r cos A
Yrel = r sin A
A = 90 + (360*3)t
A = 90 + 1080 t
cos A = cos (90 + 1080t)
= -sin(1080 t) trig identity
sin A = cos(1080 t)
so
Xrel = -1.5 sin (1080 t)
Yrel = 1.5 cos (1080 t)
and sum the x and y components
xpoint = 17 t -1.5 sin (1080 t)
ypoint = 17 t +1.5 cos (1080 t)
you can do conversion to polar coordinates I am sure if needed.
Note my angles are in degrees, not radians
To answer these questions, we can use the concept of circular motion and trigonometry. Let's break it down step by step:
(a) The center of the puck is traveling at a constant speed in circular motion. We are given the diameter of the puck, which is 3 inches. The circumference of a circle is given by 2πr, where r is the radius. In this case, the radius is half the diameter, so r = 3/2 = 1.5 inches.
Since the puck is spinning counterclockwise, we can say its angular speed (ω) is 3 rotations per second. The angular speed is the rate at which the angle changes as the puck spins. In this case, the angle changes by 2π radians in one rotation.
To find the angular displacement (θ) at a given time t, we multiply the angular speed by t. So, θ = ωt = 3 * t * 2π.
Now, let's find the x and y coordinates of the center of the puck at time t.
The x-coordinate can be found by using the formula for circular motion along the x-axis: x = r * cos(θ).
Substituting the values: x = (1.5) * cos(3 * t * 2π).
The y-coordinate can be found by using the formula for circular motion along the y-axis: y = r * sin(θ).
Substituting the values: y = (1.5) * sin(3 * t * 2π).
(b) To find the location of a point on the outer edge of the puck at time t, we need to consider its initial position and the angular displacement.
The initial position of the point is given as (0, 3/2).
We can assume the initial angle of the point is 0 radians (from the positive x-axis).
To find the current angle (θ) of the point, we add the initial angle to the angular displacement of the center of the puck: θ = 3 * t * 2π.
Now, let's find the x and y coordinates of the point at time t.
The x-coordinate can be found by using the formula for circular motion along the x-axis: x = r * cos(θ).
Substituting the values: x = (1.5) * cos(3 * t * 2π).
The y-coordinate can be found by using the formula for circular motion along the y-axis: y = r * sin(θ).
Substituting the values: y = (1.5) * sin(3 * t * 2π).
So, to summarize:
(a) The center of the puck is located at (x, y) = (1.5 * cos(3 * t * 2π), 1.5 * sin(3 * t * 2π)).
(b) A point on the outer edge of the puck is located at (x, y) = (1.5 * cos(3 * t * 2π), 1.5 * sin(3 * t * 2π)) + (0, 3/2).