3. A length of rope is used to pull a mass $m$ block along a horizontal surface as shown. The tension $\vec{T}$ . in the rope is applied at an angle $ϕ$ above the horizontal, the horizontal surface has coefficient of kinetic friction $μ_{k}$ with the block as it slides, and the block has acceleration $\vec{a}$ . Which expression gives the magnitude of the tension, T? Assume the acceleration due to gravity is $a_{y} = - g$ .
A. $\frac{m a}{\cos ϕ}$
B. $\frac{μ_{k} m a}{\tan ϕ}$
C. $\frac{m (a + μ_{k} g)}{\cos ϕ + μ_{k} \sin ϕ}$
D. $\frac{m a}{\cos ϕ - μ_{k} \sin ϕ}$ .

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Answer

Combine the terms to get the final expression for T: $T = \frac{m (a + μ_{k} g)}{\cos (ϕ) + μ_{k} \sin (ϕ)}$

Steps

Step 1 ：Break the tension T into its horizontal and vertical components: $T_{x} = T \cos (ϕ)$ and $T_{y} = T \sin (ϕ)$

Step 2 ：Write Newton's second law equations for horizontal and vertical forces: $- N μ_{k} + T \cos (ϕ) = m a$ and $N + T \sin (ϕ) - m g = 0$

Step 3 ：Solve the system of equations for T: $T = \frac{m a}{\cos (ϕ) + μ_{k} \sin (ϕ)} + \frac{m g μ_{k}}{\cos (ϕ) + μ_{k} \sin (ϕ)}$

Step 4 ：Combine the terms to get the final expression for T: $T = \frac{m (a + μ_{k} g)}{\cos (ϕ) + μ_{k} \sin (ϕ)}$