The connecting rod from the piston to the crankshaft in a certain 2.0 L engine is 6.4 in. The radius of the crank circle is 2.8 in. If the angle made by the connecting rod with the horizontal at the wrist pin P is 20 degrees, how far is the wrist pin from the center C of the crankshaft? Round to the nearest tenth of an inch.
To find the distance between the wrist pin P and the center of the crankshaft C, we can use trigonometry.
First, let's draw a diagram to visualize the problem:
P
*|\
R *| \
* *| \
* θ *| \ 6.4in
* *| \
* *| \
* *| \
* *| \
* *| \
C__________*________*_______
We can see that the distance between P and C is the hypotenuse of a right triangle formed by the connecting rod (6.4 in) and the horizontal distance between P and C.
Using trigonometry, we know that:
cos(θ) = adjacent / hypotenuse
We can rearrange this equation to solve for the adjacent side:
adjacent = cos(θ) * hypotenuse
adjacent = cos(20°) * 6.4 in
adjacent ≈ 6.4 in * 0.9397
adjacent ≈ 6.03968 in
Therefore, the wrist pin is approximately 6.0 inches away from the center of the crankshaft.
To find the distance between the wrist pin and the center of the crankshaft, we can use the law of sines.
Let's denote the distance between the wrist pin (P) and the center of the crankshaft (C) as x.
By the law of sines, we have:
sin(20°)/6.4 = sin(90° - 20°)/x
Simplifying this equation, we get:
sin(20°)/6.4 = cos(20°)/x
Now, let's solve for x:
x = (cos(20°) * 6.4) / sin(20°)
Using a calculator, we find:
x ≈ (0.9397 * 6.4) / 0.3420
x ≈ 17.66 / 0.3420
x ≈ 51.56
Therefore, the distance between the wrist pin and the center of the crankshaft is approximately 51.6 inches.
To determine the distance from the wrist pin P to the center C of the crankshaft, we can use the law of cosines. The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the two sides and the cosine of their included angle.
In this case, we have a triangle with sides:
- The length of the connecting rod, which is given as 6.4 inches
- The radius of the crank circle, which is given as 2.8 inches
- The distance from the wrist pin P to the center C of the crankshaft, which we need to find
The included angle between the connecting rod and the radius of the crank circle is given as 20 degrees.
Therefore, we can use the law of cosines to solve for the unknown side (distance from P to C):
distance from P to C^2 = 6.4^2 + 2.8^2 - 2 * 6.4 * 2.8 * cos(20)
Now, let's calculate the value using the formula:
distance from P to C^2 = 40.96 + 7.84 - (2 * 6.4 * 2.8 * cos(20))
To find the square root of this value, we can use a calculator or a programming language. Thus, the distance from the wrist pin P to the center C of the crankshaft is approximately 5.1 inches.