If a circle of radius 5 is made to roll along the x-axis, what is the equation for the path of the center of the circle?
By definition of a circle, the centre is always at the same distance (=radius) from the circumference, or more precisely, any tangent to the circle.
If the circle of radius 5 rolls above the x-axis, the locus of the centre is therefore y=5. If it rolls under the x-axis, the locus of the centre is y=-5.
Well, the center of the circle will be moving in a horizontal line along the x-axis. So the equation for the path of the center of the circle would simply be x = t, where t represents time. But hey, if you're looking for a more clownish answer, I could say the center will be doing the Macarena while rolling on the x-axis. The equation for that might be a little harder to pin down though! 😅
To determine the equation for the path of the center of the circle, we first need to understand the motion of the circle as it rolls along the x-axis.
As the circle rolls, its center moves horizontally while staying a fixed distance from the x-axis. This means that the x-coordinate of the center of the circle will change, while the y-coordinate remains constant.
Let's assume the center of the circle starts at the point (0, 5) on the coordinate plane.
As the circle rolls, it completes one full revolution after traveling a distance equal to the circumference of the circle.
The circumference of a circle can be found using the formula: C = 2Ï€r, where r is the radius.
For this circle with a radius of 5, the circumference is C = 2Ï€(5) = 10Ï€.
So, after one full revolution, the center of the circle would have traveled a distance of 10Ï€ along the x-axis.
Therefore, the equation for the path of the center of the circle is:
x = 10Ï€t
where x represents the x-coordinate of the center of the circle at any given time t.
Note: This equation assumes the circle starts rolling from the point (0, 5). If it starts from a different point, the equation would need to be adjusted accordingly.
To determine the equation for the path of the center of the circle as it rolls along the x-axis, you need to consider the motion of the circle. As it rolls, the center of the circle traces a curve called a cycloid.
The equation for a cycloid with a radius of 5 can be expressed as:
x = r(θ - sinθ)
y = r(1 - cosθ)
Where:
- x and y are the coordinates of the center of the circle at a particular angle θ,
- r is the radius of the circle (in this case, 5),
- sin and cos are the trigonometric functions sine and cosine respectively.
Since the circle is rolling along the x-axis, the initial position of the center of the circle is (5, 0) (when θ = 0).
Hence, the equation for the path of the center of the circle as it rolls along the x-axis would be:
x = 5(θ - sinθ)
y = 5(1 - cosθ)