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Alex777 [14]
1 year ago
11

The coordinates of the midpoint of MN are (4,1). The coordinates of point M are (-2,5).What are the coordinates of point N?

Mathematics
1 answer:
Ilya [14]1 year ago
8 0

The Solution:

Step 1: The given data

The coordinates of the midpoint of MN is (4,1)

The coordinates of point of M is (-2,5)

The coordinates of point of N is (x,y)

Step 2:

The formula for calculating Midpoint coordinates is given below:

\text{Midpoint of MN=(}\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})=(4,1)

In this case,

\begin{gathered} x_1=-2,y_1=5 \\ x_2=x,y_2=y \end{gathered}

Substituting these values in the formula above, we get

\begin{gathered} 4\text{=}\frac{-2_{}+x_{}}{2} \\ \text{cross multiplying, we get} \\ -2+x=4\times2 \end{gathered}\begin{gathered} -2+x=8 \\ x=8+2=10 \end{gathered}

Similarly,

\begin{gathered} 1\text{=}\frac{5_{}+y_{}}{2} \\ \text{cross multiplying, we get} \\ 5+y=2\times1 \end{gathered}y=2-5=-3

Hence, the coordinates of point N is (10,-3)

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Does anyone know the answer?
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Answer:

a. θ = 30°

b. μ = √3 / 15 ≈ 0.115

Step-by-step explanation:

Draw a free body diagram for each scenario (see attached figure).  The body has four forces acting on it:

  • Weight pulling down
  • Normal force perpendicular to the incline
  • Applied force parallel up the incline
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Remember that friction opposes the direction of motion.  So when the body is sliding up, friction points down the incline.  And when the body is sliding down, friction points up the incline.

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Sum of the forces parallel to the incline:

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Substituting the expression for normal force:

P₁ − mgμ cos θ − mg sin θ = 0

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And sum of the forces parallel to the incline:

∑F = ma

P₂ + f − mg sin θ = 0

P₂ + Nμ − mg sin θ = 0

Substituting the expression for normal force:

P₂ + mgμ cos θ − mg sin θ = 0

We know that P₁ = 6 kg.wt, P₂ = 4 kg.wt, and mg = 10 kg.wt.

So we have two equations and two unknowns (μ and θ):

P₁ − mgμ cos θ − mg sin θ = 0

P₂ + mgμ cos θ − mg sin θ = 0

Let's start by adding the equations together:

P₁ + P₂ − 2 mg sin θ = 0

P₁ + P₂ = 2 mg sin θ

sin θ = (P₁ + P₂) / (2 mg)

Plugging in the values:

sin θ = (6 + 4) / (2 × 10)

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Now we can plug this into either equation and find μ.

P₁ − mgμ cos θ − mg sin θ = 0

6 − (10 cos 30°) μ − 10 sin 30° = 0

6 − 5√3 μ − 5 = 0

1 = 5√3 μ

μ = √3 / 15

μ ≈ 0.115

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