a seesaw consists of a plank 4.5m long which is supported by a pivot at its center and moves in a vertical plane above the pivot. if the height of the pivot pillar above the ground is 1m, through what maximum angle can the seesaw beam move?
we could have just used the cosine law
1^2 = 4.5^2 + 4.5^2 - 2(4.5)(4.5)cosØ
1 = 20.25 + 20.25 - 40.5cosØ
40.5 cosØ = 39.5
cosØ = 39.5/40.5 = .9753
Ø = 12.759°
And I just realized that in my first solution I should have used the sine instead of tangent
sin QFP = .5/4.5 = 5/45 = 1/9
angle QFP = appr 6.3794°
So the beam moves through an angle of 12.759°
just like above
Did you make a sketch ?
I see an isosceles triangle on its side with equal sides 4.5 and a vertical line of 1
label the top of the plank P, the fulcrum F
draw a horizontal to hit the height at Q
PFQ is a right-angled triangle with hypotenuse 4.5, and a vertical leg of .5
angle QFP is half of the angle we want.
tan QFP = .5/4.5 = 5/45 = 1/9
angle QFP = appr 6.34°
So the beam moves through an angle of 12.68°
To determine the maximum angle the seesaw beam can move, we can use trigonometry.
In this case, we have a right triangle where the height of the pivot pillar (opposite side) is 1m and the length of the plank (hypotenuse) is 4.5m. We need to find the angle (θ) that corresponds to the maximum height.
We can use the inverse sine function (sin^-1) to solve for the angle:
sin^-1(opposite / hypotenuse) = sin^-1(1/4.5)
Using a calculator, we can determine:
sin^-1(1/4.5) ≈ 12.182 degrees
Therefore, the maximum angle the seesaw beam can move is approximately 12.182 degrees.
To find the maximum angle through which the seesaw can move, we need to consider the height of the seesaw's end when it is at its highest position.
Given that the plank is 4.5 meters long and pivots at its center, the distance from the pivot to one end of the seesaw is half of its length, which is 4.5 / 2 = 2.25 meters.
Now, let's create a right triangle with the seesaw as the hypotenuse and the vertical height above the ground as one leg. The other leg will be the horizontal distance from the pivot to the end of the seesaw.
Using the Pythagorean theorem, we can find this horizontal distance:
horizontal distance = √(seesaw length^2 - vertical height^2)
horizontal distance = √(2.25^2 - 1^2)
horizontal distance = √(5.0625 - 1)
horizontal distance = √4.0625
horizontal distance ≈ 2.02 meters
Now that we have the horizontal distance, we can calculate the maximum angle using trigonometry. The sine function will be used as follows:
sin(angle) = vertical height / seesaw length
In this case, we want the inverse of the sine function (sin^-1) to get the angle:
angle = sin^-1(vertical height / seesaw length)
angle = sin^-1(1 / 2.25)
angle ≈ sin^-1(0.44)
angle ≈ 26 degrees
Therefore, the maximum angle through which the seesaw can move is approximately 26 degrees.