A wheel with radius 5 cm is being pushed up a ramp at a rate of 6 cm per second. The ramp is 540 cm long, and 260 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 8 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 5 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. To find the time it takes for the wheel to complete one rotation, we can use the formula:
Time = Distance / Speed
The distance covered in one rotation is equal to the circumference of the wheel, which is given by 2πr, where r is the radius of the wheel. In this case, the radius is 5 cm.
So, the distance covered in one rotation is 2π * 5 cm.
Given that the speed of the wheel is 8 cm/sec, we can substitute the values into the formula:
Time = (2π * 5 cm) / 8 cm/sec
Simplifying, we get:
Time = (10π cm) / 8 cm/sec
Therefore, it takes (10π / 8) seconds for the wheel to complete one rotation.
2. The initial height of the center of the wheel is the starting point on the ramp. Since the ramp is inclined at an angle, the initial height can be calculated using trigonometry.
Given that the ramp is 260 cm tall at the end, and the length of the ramp is 540 cm, we can calculate the height using the tangent function:
Initial height = 260 cm * cos(arctan(260 cm / 540 cm))
Simplifying, we get:
Initial height = 260 cm * cos(0.456)
Calculating the value, the initial height is approximately 207.5 cm.
3. To find how much the height of the center of the wheel increases when the wheel travels at a speed of 5 cm/sec for 1 second, we can use a similar approach.
Since the wheel is moving horizontally along the ramp, the increase in height is equal to the vertical component of the motion, which can be calculated using trigonometry.
Given that the speed is 5 cm/sec and the ramp is inclined at an angle, we can calculate the increase in height using the sine function:
Increase in height = 5 cm * sin(arctan(260 cm / 540 cm))
Simplifying, we get:
Increase in height = 5 cm * sin(0.456)
Calculating the value, the height of the center of the wheel increases by approximately 2.28 cm.
4. To write an equation for the height of the center of the wheel as a function of time (t) in seconds, we need to consider the vertical component of the motion along the ramp.
Since the ramp is inclined at an angle, we can express the height (h) at time t using the tangent function:
h(t) = Initial height + t * speed * sin(arctan(260 cm / 540 cm))
Substituting the values, the equation becomes:
h(t) = 207.5 cm + t * speed * sin(0.456)
5. To write an equation for the height of the point P above the ground as a function of time (t) in seconds, we need to consider the height of the center of the wheel (h(t)) and the radius of the wheel (5 cm).
The height of point P is given by:
Height of P = h(t) + 5 cm
Substituting the equation for h(t) we obtained earlier, the equation becomes:
Height of P = 207.5 cm + t * speed * sin(0.456) + 5 cm
Simplifying, we get:
Height of P = 212.5 cm + t * speed * sin(0.456)
To find the answers to these questions, we need to break down the problem and apply some concepts of geometry and physics.
1. To find the time it takes for the wheel to complete one rotation at a speed of 8 cm/sec, we can use the formula: time = distance / speed.
The distance traveled in one rotation is equal to the circumference of the wheel, which can be found using the formula: circumference = 2 * π * radius.
Given that the radius of the wheel is 5 cm, the circumference is: circumference = 2 * π * 5 = 10π cm.
Now, we can calculate the time it takes to complete one rotation: time = 10π / 8 ≈ 3.93 seconds.
So, it takes approximately 3.93 seconds for the wheel to complete one rotation at a speed of 8 cm/sec.
2. The initial height of the center of the wheel can be found by considering the geometry of the problem. The wheel is being pushed up a ramp, and we are given the dimensions of the ramp.
The ramp is 260 cm tall at the end, and the wheel is being pushed up at a rate of 6 cm/sec. This means that the wheel will take 260 / 6 ≈ 43.33 seconds to reach the top of the ramp.
Since we want to determine the initial height, we need to subtract the distance traveled on the ramp from the total height of the ramp.
Initial height = total height - distance traveled on the ramp
Initial height = 260 cm - (540 cm - (6 cm/sec * 43.33 sec))
Initial height = 260 cm - (540 cm - 260 cm)
Initial height = 260 cm - 280 cm
Initial height = -20 cm
Therefore, the initial height of the center of the wheel is -20 cm. Note that the negative sign indicates that the wheel is initially below the ground level.
3. To find how much the height of the center of the wheel increases in one second at a speed of 5 cm/sec, we need to consider the rate of change of height.
Given that the wheel is traveling at 5 cm/sec, we can use this value as the derivative of the height function.
Thus, the height of the center of the wheel increases at a rate of 5 cm/sec.
4. Writing an equation for the height of the center of the wheel as a function of time (t) in seconds would be:
Height = initial height + (rate of change of height * time)
Height = -20 cm + (5 cm/sec * t)
Height = -20 cm + 5t cm
Therefore, the equation for the height of the center of the wheel as a function of time (t) is Height = -20 cm + 5t cm.
5. To write an equation for the height of point P above the ground as a function of time (t) in seconds, we need to consider the geometry of the problem.
Given that point P is marked on the circle, its height above the ground will depend on both the height of the center of the wheel and the radius of the wheel.
The height of point P can be determined by adding the height of the center of the wheel to the vertical distance from the center of the wheel to point P, which is equal to the radius of the wheel.
Therefore, the equation for the height of point P above the ground as a function of time (t) is:
Height of P = height of center + radius
Height of P = (-20 cm + 5t cm) + 5 cm
Height of P = -20 cm + 5t cm + 5 cm
Height of P = -20 cm + 5t + 5 cm
Height of P = -15 cm + 5t cm
Hence, the equation for the height of point P above the ground as a function of time (t) is Height of P = -15 cm + 5t cm.
It's kind of an oddly worded question - it states a speed of 6cm/sec, but goes on immediately in part 1. to change this to 8cm/s...
1. A point on the wheel travels a distance of the circumference of the wheel in one rotation, or 2π * 5 = 10π cm
10π cm divided by 8 cm/s = _____s
2. You should be able to work this out from the diagram (which can't be seen on this forum). As a hint, if the wheel were on level ground, the height of the centre of the wheel above the ground would be the wheel's radius.
3. The centre of the wheel travels 5 cm in 1 second, in a direction parallel to the ramp's surface. The angle the ramp makes with the ground is θ = tan-1(ramp slope).
So the vertical component of the centre of the wheel's displacement (the height increase) would be 5sinθ cm.
This should be enough info to get you going on the rest of the question.