Two blocks, with masses m=3.6 kg and M=7.1 kg, are connected by a lightweight wire that's shown in the picture. The incline makes an angle of 55 degrees with the horizontal and the coefficient of friction between each block and the surface is 0.200. Find a) the acceleration of the blocks and b) the tension in the string. The image depicts a simple mechanical system involving two blocks and an inclined plane. - **Block $ m $:** Positioned on a horizontal surface. - **Block $ M $:** Located on an inclined plane, connected to block $ m $ via a pulley system.- **Inclined Plane:** The slope on which block $ M $ rests.This setup is commonly used in physics problems to illustrate concepts such as tension in the connecting string, gravitational force components, and how angles affect mechanical advantage. The system can be analyzed using Newton's laws to determine forces acting on each block.

Given data: m=3.6 kg M=7.1 kg θ=55O Coefficient of friction, μ=0.200 Applying Newton’s second…

Answered: Two blocks, with masses m=3.6 kg and…

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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Two blocks, with masses m=3.6 kg and M=7.1 kg, are connected by a lightweight wire that's shown in the picture. The incline makes an angle of 55 degrees with the horizontal and the coefficient of friction between each block and the surface is 0.200. Find a) the acceleration of the blocks and b) the tension in the string.

The image depicts a simple mechanical system involving two blocks and an inclined plane.

- **Block $ m $:** Positioned on a horizontal surface.
- **Block $ M $:** Located on an inclined plane, connected to block $ m $ via a pulley system.
- **Inclined Plane:** The slope on which block $ M $ rests.

This setup is commonly used in physics problems to illustrate concepts such as tension in the connecting string, gravitational force components, and how angles affect mechanical advantage. The system can be analyzed using Newton's laws to determine forces acting on each block.

Expert Solution

Step 1

Given data:

$\begin{array}{rcl} m & = & 3.6 kg \\ M & = & 7.1 kg \\ θ {=55}^{O} \end{array}$

Coefficient of friction, $μ = 0.200$

Applying Newton’s second law on the block on horizontal surface:

$\begin{array}{rcl} T - f_{1} & = & m a \\ T - μ N_{1} & = & m a \\ T - μ m g & = & m a \\ T - 0.200 \times 3.6 \times 9.8 & = & 3.6 a \\ T - 7.056 & = & 3.6 a . . . . . . . . . . . . . (i) \end{array}$

Applying Newton’ second law on the block on inclined surface:

$\begin{array}{rcl} M g \sin θ - T - f_{2} & = & M a \\ M g \sin θ - T - μ N_{2} & = & M a \\ M g \sin θ - T - μ M g \cos θ & = & M a \\ 7.1 \times 9.8 \times \sin 55^{o} - T - 0.200 \times 7.1 \times 9.8 \times \cos 55^{o} & = & 7.1 a \\ 56.99 - T - 7.98 & = & 7.1 a \\ 49.01 - T & = & 7.1 a . . . . . . . (ii) \end{array}$

(a)

Adding equation (i) and (ii):

$\begin{array}{rcl} 49.01 - 7.056 & = & 7.1 a + 3.6 a \\ 41.95 & = & 10.7 a \\ a & = & 3.92 {m/s}^{2} \end{array}$

(b)

Plugging in the value of acceleration in equation (i):

$\begin{array}{rcl} T - 7.056 & = & 3.6 \times 3.92 \\ T & = & 21.17 N \end{array}$

Answer:

(a) Acceleration: 3.92 m/s²

(b) Tension in the rope: 21.17 N

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