Math Snap

STEP 1

1. The coefficient of kinetic friction between $m_{\mathrm{A}}$ and the ramp is $\mu_{\mathrm{k}} = 0.15$.
2. Both masses $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ are equal to $2.7 \, \text{kg}$.
3. The angle of the ramp $\theta$ is $34^{\circ}$.
4. The pulley is frictionless, and the masses of the cord and pulley are ignored.
5. We need to find the acceleration of the system and the smallest value of $\mu_{\mathrm{k}}$ that will keep the system from accelerating.

STEP 2

1. Analyze forces acting on the masses.
2. Write equations of motion for the system.
3. Solve for the acceleration of the system.
4. Determine the smallest value of $\mu_{\mathrm{k}}$ to prevent acceleration.

STEP 3

Analyze forces acting on the masses.
For $m_{\mathrm{A}}$ on the ramp:
- Gravitational force component along the ramp: $m_{\mathrm{A}} g \sin \theta$
- Normal force: $N = m_{\mathrm{A}} g \cos \theta$
- Frictional force: $f_{\mathrm{k}} = \mu_{\mathrm{k}} N = \mu_{\mathrm{k}} m_{\mathrm{A}} g \cos \theta$
For $m_{\mathrm{B}}$:
- Gravitational force: $m_{\mathrm{B}} g$

STEP 4

Write equations of motion for the system.
For $m_{\mathrm{A}}$ moving up the ramp:
$$T - m_{\mathrm{A}} g \sin \theta - f_{\mathrm{k}} = m_{\mathrm{A}} a$$ For $m_{\mathrm{B}}$ moving down:
$$m_{\mathrm{B}} g - T = m_{\mathrm{B}} a$$

STEP 5

Solve for the acceleration of the system.
Add the two equations to eliminate $T$:
$$m_{\mathrm{B}} g - m_{\mathrm{A}} g \sin \theta - \mu_{\mathrm{k}} m_{\mathrm{A}} g \cos \theta = (m_{\mathrm{A}} + m_{\mathrm{B}}) a$$ Solve for $a$:
$$a = \frac{m_{\mathrm{B}} g - m_{\mathrm{A}} g \sin \theta - \mu_{\mathrm{k}} m_{\mathrm{A}} g \cos \theta}{m_{\mathrm{A}} + m_{\mathrm{B}}}$$ Substitute the given values:
$$a = \frac{2.7 \times 9.8 - 2.7 \times 9.8 \times \sin 34^{\circ} - 0.15 \times 2.7 \times 9.8 \times \cos 34^{\circ}}{2.7 + 2.7}$$ Calculate $a$.

Determine the smallest value of $\mu_{\mathrm{k}}$ to prevent acceleration.
Set $a = 0$ in the equation:
$$m_{\mathrm{B}} g = m_{\mathrm{A}} g \sin \theta + \mu_{\mathrm{k}} m_{\mathrm{A}} g \cos \theta$$ Solve for $\mu_{\mathrm{k}}$:
$$\mu_{\mathrm{k}} = \frac{m_{\mathrm{B}} g - m_{\mathrm{A}} g \sin \theta}{m_{\mathrm{A}} g \cos \theta}$$ Substitute the given values:
$$\mu_{\mathrm{k}} = \frac{2.7 \times 9.8 - 2.7 \times 9.8 \times \sin 34^{\circ}}{2.7 \times 9.8 \times \cos 34^{\circ}}$$ Calculate $\mu_{\mathrm{k}}$.