A wheel with a radius of 3 cm is being pushed up a ramp at a rate of 7 cm per second. The ramp is 640 cm long, and 200 cm tall at the end. A point P is marked on the circle as shown (picture is not to scale).
wheelonramp.gif
1. Traveling at 7 cm/sec, how long does it take the wheel to complete one rotation?
2. What is the initial height of the center of wheel?
3. Traveling at 7 cm/sec, in one second, how much does the height of the center of wheel increase?
4. Write an equation for the height of the center of the wheel as a function of t, time in seconds.
5. Write an equation for the height of the point P above the ground as a function of t, time in seconds.
#1. C = 6π cm, so it takes 6π/7 = 2.69 seconds to rotate once
#2. No idea where P is, but the plane is inclined at an angle of
Ø = arctan(200/640) = 17.35°
So, if P is such that the radius from the center of the wheel is perpendicular to the ramp, and the bottom of the ramp is at (0,0), then P = (-3sinØ,3cosØ)
#3. the center travels vertically at 7sinØ cm/s
#4. should be obvious
#5. see what you can do here. Without knowing where P started, I don't want to bother doing the calculation. But you know that the angle θ grows by w=2πf = 2π * 7/(6π) = 7/3 rad/s
1. To find out how long it takes for the wheel to complete one rotation, we need to determine the circumference of the wheel. The circumference of a circle can be calculated using the formula C = 2πr, where C is the circumference and r is the radius of the circle.
In this case, the radius of the wheel is given as 3 cm. So, the circumference of the wheel is 2π(3) = 6π cm.
Since the wheel is traveling at a rate of 7 cm per second, we can now determine the time it takes to complete one rotation by dividing the circumference of the wheel by the rate of travel.
Time = Circumference / Rate = 6π / 7 ≈ 2.72 seconds
Therefore, it takes approximately 2.72 seconds for the wheel to complete one rotation.
2. The initial height of the center of the wheel can be determined by examining the given dimensions of the ramp. We know that the ramp is 200 cm tall at the end and the wheel is being pushed up the ramp.
Since the wheel is being pushed up the ramp, its center must start at a lower height than the height of the ramp. Therefore, the initial height of the center of the wheel can be considered as 0 cm.
3. To determine how much the height of the center of the wheel increases in one second while traveling at a rate of 7 cm per second, we need to consider the ratio between the vertical distance traveled and the distance traveled along the ramp.
The ramp is 640 cm long and the wheel is moving up the ramp at a rate of 7 cm per second. Therefore, in one second, the wheel travels horizontally a distance of 7 cm.
Using similar triangles, we can calculate the corresponding vertical distance traveled by the center of the wheel.
Vertical Distance = (Vertical Height of Ramp / Horizontal Distance of Ramp) * Distance Traveled Horizontal
= (200 cm / 640 cm) * 7 cm
= 2.1875 cm
Therefore, in one second, the height of the center of the wheel increases by approximately 2.1875 cm.
4. The equation for the height of the center of the wheel as a function of time (t) can be represented as h(t) = mt + b, where m is the rate of increase in height and b is the initial height.
From the previous calculations, we found that the rate of increase in height is 2.1875 cm per second and the initial height is 0 cm.
Therefore, the equation for the height of the center of the wheel as a function of time (t) is h(t) = 2.1875t + 0, which simplifies to h(t) = 2.1875t.
5. To write an equation for the height of point P above the ground as a function of time (t), we need to consider the fact that as the wheel rotates, point P on the wheel goes up and down.
The height of point P above the ground can be determined by adding the height of the center of the wheel (h(t)) to the radius of the wheel.
Therefore, the equation for the height of point P above the ground as a function of time (t) can be represented as P(t) = h(t) + r, where h(t) is the height of the center of the wheel and r is the radius of the wheel (3 cm).
Substituting the value of h(t) from the previous equation, we get P(t) = 2.1875t + 3.
Thus, the equation for the height of point P above the ground as a function of time (t) is P(t) = 2.1875t + 3.