Question

homework 3 begin date: 8/17/2025 11:59:00 pm due date: 9/9/2025 11:59:00 pm end date: 9/9/2025 11:59:00 pm problem 16: (6% of assignment value) two blocks are connected by an ideal string which passes over an ideal pulley which is massless and frictionless, as shown. the inclination of the ramp is θ = 43.3°, while the masses of the blocks are m₁ = 6.7 kg and m₂ = 20.9 kg. friction is negligible. part (a) enter an expression for the magnitude of the acceleration of the two blocks. give your equation in terms of the variables defined in the problem statement, and the use g for the acceleration due to gravity. assume that block 2 descends vertically.

Explanation:

Step1: Analyze forces on each block

For block \( m_2 \) (descending vertically), the net force is \( F_{net2}=m_2g - T \), where \( T \) is the tension in the string. By Newton's second law, \( m_2g - T=m_2a \).
For block \( m_1 \) (on the ramp), the component of gravity along the ramp is \( m_1g\sin\theta \), and the net force is \( F_{net1}=T - m_1g\sin\theta \). By Newton's second law, \( T - m_1g\sin\theta=m_1a \).

Step2: Eliminate tension \( T \)

From the first equation, \( T = m_2g - m_2a \). Substitute this into the second equation:
\( m_2g - m_2a - m_1g\sin\theta=m_1a \).

Step3: Solve for acceleration \( a \)

Rearrange the equation:
\( m_2g - m_1g\sin\theta=m_1a + m_2a \)
Factor out \( a \) on the right:
\( m_2g - m_1g\sin\theta=a(m_1 + m_2) \)
Divide both sides by \( (m_1 + m_2) \):
\( a=\frac{m_2g - m_1g\sin\theta}{m_1 + m_2} \) or factor \( g \): \( a = g\frac{m_2 - m_1\sin\theta}{m_1 + m_2} \)

Answer:

\( \boldsymbol{\frac{m_2 g - m_1 g \sin\theta}{m_1 + m_2}} \) (or equivalent factored form \( g\frac{m_2 - m_1\sin\theta}{m_1 + m_2} \))