A wheel with radius 5 cm is being pushed up a ramp at a rate of 8 cm per second. The ramp is 500 cm long, and 280 cm tall at the end. A point P is marked on the circle as shown (picture is not to scale).
Cross-sectional diagram of a ramp with a wheel at the base. The shape is a right triangle, and the hypotenuse and vertical sides are labeled with the given length and height. Point P is at the topmost point of the wheel.
500 cm280 cm
Write parametric equations for the position of the point P as a function of t, time in seconds after the ball starts rolling up the ramp. Both
x
and
y
are measured in centimeters.
x
=
y
=
when the wheel has rotated through an angle θ, then if the center of the circle is at (h,k) we have
x = h+5cosθ
y = k-5sinθ
so the real trick is, what are θ,h,k in terms of time t?
dθ = 10πcm/8s = 5π/4 rad/s
so θ = 5π/4 t (it starts with θ(0) = 0)
The ramp makes an angle Ø such that tanØ = 280/500
When the wheel starts to roll up the plank, the radius perpendicular to the plank has location (h,k) where
h = -5sinØ
k = 5cosØ
and the horizontal and vertical speeds are
8cosØ and 8sinØ
when you put that all together, you might consider comparing it to the equations of a cycloid:
x = r(θ-sinθ)
y = r(1-cosθ)
adjusted for the sloping ramp.
If you get stuck, come on back with what you got.
my answer got wrong and I got new one still couldn't figure out the right answer
A wheel with a radius 2 cm is being pushed up a ramp at a rate of 7 cm per second. The ramp is 700 cm long, and 140 cm tall at the end. A point P is marked on the circle as shown (picture is not to scale).
Cross-sectional diagram of a ramp with a wheel at the base. The shape is a right triangle, and the hypotenuse and vertical sides are labeled with the given length and height. Point P is at the topmost point of the wheel.
700 cm140 cm
Write parametric equations for the position of the point P as a function of t, time in seconds after the ball starts rolling up the ramp. Both
x
and
y
are measured in centimeters.
x
=
y
=
You will have a radical expression for part of the horizontal component. It's best to use the exact radical expression even though the answer that WAMAP shows will have a decimal approximation.
To write parametric equations for the position of point P on the wheel, we need to consider the motion of the wheel along the ramp.
Let's consider the triangle formed by the ramp, where the base has a length of 500 cm and the height is 280 cm. The hypotenuse of this triangle represents the path of the wheel as it moves up the ramp.
To find the equation for the x-coordinate of point P, we can use the concept of similar triangles. The ratio of the horizontal distance traveled by the wheel (x) to the length of the ramp (500 cm) is equal to the ratio of the radius of the wheel (5 cm) to the height of the ramp (280 cm).
Therefore, the equation for the x-coordinate of point P can be written as:
x = (5 cm / 280 cm) * t
Simplifying this equation gives us:
x = t / 56 cm
Similarly, to find the equation for the y-coordinate of point P, we can again use the concept of similar triangles. The ratio of the vertical distance traveled by the wheel (y) to the length of the ramp (500 cm) is equal to the ratio of the radius of the wheel (5 cm) to the height of the ramp (280 cm).
Therefore, the equation for the y-coordinate of point P can be written as:
y = (5 cm / 280 cm) * t
Simplifying this equation gives us:
y = t / 56 cm
So, the parametric equations for the position of point P as a function of time (t) are:
x = t / 56 cm
y = t / 56 cm