Question

Susan's 11.0 kg baby brother Paul sits on a mat. Susan pulls the mat across the floor using a rope that is angled 30∘ above the floor. The tension is a constant 30.0 N and the coefficient of friction is 0.180.

Use work and energy to find Paul's speed after being pulled 3.10 m .

Answer 1

Answer on Question #63514, Physics / Mechanics | Relativity

Question:

Susan's $11.0\mathrm{kg}$ baby brother Paul sits on a mat. Susan pulls the mat across the floor using a rope that is angled $30{}^{\circ}$ above the floor. The tension is a constant $30.0\mathrm{N}$ and the coefficient of friction is 0.180.

Use work and energy to find Paul's speed after being pulled $3.10\mathrm{m}$

Solution:

Let us assume that the weight of the mat is negligibly small. To use work and energy we must consider horizontal components of forces acting on Paul.

$$F _ {T} ^ {x} = F _ {T} \cdot \cos \alpha$$

$F_{F}^{x} = kF_{N} = kmg$ , where $k$ is the coefficient of friction, $m$ is the mass of Paul and $g$ is the gravitational acceleration.

The resultant force $F_{res} = F_T \cdot \cos \alpha - kmg$

The work of this force $W = (F_T \cdot \cos \alpha - kmg) \cdot s$

Kinetic energy at the point $P_{1}$ is calculated as $E_{k} = \frac{mv^{2}}{2}$ , and according to the law of conservation of energy $W = E_{k}$ or $(F_{T} \cdot \cos \alpha - kmg) \cdot s = \frac{mv^{2}}{2}$ .

Then the speed $v = \sqrt{\frac{2s\cdot(F_T\cdot\cos\alpha - kmg)}{m}}$ .

$$v = \sqrt {\frac {2 \cdot 3 . 1 \cdot (3 0 . 0 \cdot \cos 3 0 {}^ {\circ} - 0 . 1 8 \cdot 1 1 . 0 \cdot 9 . 8 1)}{1 1 . 0}} = 1. 9 m / s$$

Answer:

$$1. 9 m / s$$

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