1) The hour hand on a clock has a length of 12 cm. Let the orgin be at the center of the clock.
a) Sketch the graph of the vertical position of the tip of the hour hand versus the angle through which the hand turns for a time period of 72h. Assume that the hour hand starts at 9.
C)How many cycles appear in the graph in part a)
I have no idea how to drraw this please guide me I really need your help:( Thanks!
YEAH BUT WHY EXPLAIN WHY
Sure! I can guide you through the process. Let's start with drawing the graph of the vertical position of the tip of the hour hand versus the angle through which the hand turns.
To draw the graph, we need to understand the movement of the hour hand. The hour hand completes a full revolution (360 degrees) in 12 hours, or 720 minutes. So, for every 360 degrees or 720 minutes, the hour hand moves vertically by a height of 12 cm.
a) Sketching the graph:
1. Start by drawing the horizontal axis to represent the angle through which the hand turns. Label it with angles ranging from 0 to 360 degrees (or multiples of 360 if you prefer).
2. Draw the vertical axis to represent the vertical position of the tip of the hour hand. Label it with heights ranging from the lowest position (let's assume it starts at 0 cm) to the highest position (12 cm in this case).
3. At an angle of 0 degrees, mark the position of the hour hand at 9 o'clock, which is at the lowest point (0 cm).
4. As you move clockwise along the horizontal axis, the vertical position of the tip of the hour hand will gradually increase. At an angle of 90 degrees, mark the position of the hour hand at 12 o'clock, which is at the highest point (12 cm).
5. Continue marking key positions every 90 degrees (or multiples of 90 if you prefer), based on the known fact that the hour hand completes a full revolution in 12 hours.
6. Connect the marked positions with a smooth curve to complete the graph.
Remember to sketch the graph only for a time period of 72 hours, as mentioned in the question.
C) Determining the number of cycles:
To determine the number of cycles in the graph, you need to find how many times the hour hand completes a full revolution in the given time period.
In this case, the time period is 72 hours. Divide this by 12 (the number of hours it takes for the hour hand to complete a full revolution) to find the number of cycles.
So, 72 hours ÷ 12 hours = 6 cycles.
Therefore, the graph should have 6 cycles, showing 6 complete revolutions of the hour hand.
I hope this helps you sketch the graph and answer the question. Let me know if you need any further assistance!
Of course, I can help you with that! To sketch the graph of the vertical position of the tip of the hour hand versus the angle through which the hand turns, we first need to understand the relationship between the angle and the vertical position.
The length of the hour hand is given as 12 cm, and we can consider the hand as a line segment connecting the origin (center of the clock) to the tip of the hour hand. Since the hand starts at 9, which is three-quarters of the way around the clock, we can say that it starts at an angle of 270 degrees.
Now, let's define a coordinate system with the origin at the center of the clock and the y-axis representing the vertical position of the tip of the hour hand. The positive y-axis is pointing upwards, and the negative y-axis is pointing downwards.
As the hour hand rotates, the vertical position of its tip can be determined by the formula:
y = R * sin(θ)
where R is the length of the hour hand (12 cm) and θ is the angle through which the hand has turned.
To sketch the graph, we need to calculate the vertical position of the tip of the hour hand for different values of θ within the given time period of 72 hours.
To find the number of cycles, we need to determine how many complete cycles the graph goes through within the given time period.
One complete cycle represents a full rotation of the hour hand, which covers an angle of 360 degrees.
To calculate the number of cycles, we can divide the total angle covered within the given time period (72 hours) by 360 degrees.
Now, let's proceed step by step:
a) Sketching the graph:
1. Start by choosing intervals or values for the angle θ within the given time period. For example, you can choose intervals such as every hour or every 15 minutes to calculate the corresponding vertical positions.
2. For each value of θ, calculate the corresponding y-coordinate using the formula y = R * sin(θ), where R is the length of the hour hand (12 cm).
3. Plot the calculated values on the graph, with θ on the x-axis and y on the y-axis, to create points that represent the positions of the tip of the hour hand at different angles.
4. Connect the plotted points to form a smooth curve. The resulting graph will show the vertical position of the tip of the hour hand as it rotates through the given time period.
C) Calculating the number of cycles:
1. Determine the total angle covered within the given time period. In this case, it is 72 hours multiplied by 360 degrees per hour.
2. Divide the total angle by 360 degrees (one complete cycle) to find the number of cycles.
For example, if the total angle covered is 72 hours * 360 degrees/hour = 25,920 degrees, divide this by 360 degrees to get 72 cycles.
I hope this explanation helps you sketch the graph and calculate the number of cycles! Let me know if you have any further questions.
call your horizontal axis the t axis (usually the x-axis)
call your vertial axis the y axis
Your graph should be a sine curve, where a complete sine curve fits into 12 hours.
So label you t axis in hrs , with its end at 72
Since 72/12 = 6 , you will have to draw 6 complete sine curves
you want y = 0 when t = 9, plot that
you want y = 6 when t = 10 , plot that
you want y = 12 , when t = 12 , plot that, it will be your maximum point
you want y = 6 when t= 2
you want y = 0 when t = 3
you want y = -6 when t = 4
you want y = -12 , when t = 6, this is your minimum point.
join you points with a smooth sine curve, and repeat to t = 72+9 or 81, for 6 complete cycles of the curve.