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Suppose the angles shown in the figure are 54 ∘
 and 25 ∘. If the left-hand mass is 1.7  kg
  ,what should the right-hand mass be so that it accelerates downslope at 0.64  m/s2
 ?
Express your answer with the appropriate units.

To solve this problem, we'll break it down step by step, using Newton's laws and the given details to determine the unknown mass.

We are given the following information:
- Left-hand mass (\(m_1\)) = 1.7 kg
- Right-hand mass (\(m_2\)) = ?
- Acceleration (\(a\)) = 0.64 m/s²
- Angles: \(54^\circ\) for the left-hand incline, and \(25^\circ\) for the right-hand incline

### Step 1: Write down the force components along the slopes for both masses
For each mass, the forces acting along the incline include gravity, tension in the string, and friction (if applicable, but none is mentioned here).

#### Left-hand mass (\(m_1\)) on the \(54^\circ\) slope:
- Force due to gravity: \( F_g = m_1 g \sin(54^\circ) \)
- Tension in the string: \( T \)
- Net force: \( F_{net} = m_1 a = m_1 g \sin(54^\circ) - T \)

#### Right-hand mass (\(m_2\)) on the \(25^\circ\) slope:
- Force due to gravity: \( F_g = m_2 g \sin(25^\circ) \)
- Tension in the string: \( T \)
- Net force: \( F_{net} = m_2 a = T - m_2 g \sin(25^\circ) \)

### Step 2: Set up the system of equations

Since the tension is the same for both masses, we can set the two net forces equal but opposite in sign:
1. \( m_1 g \sin(54^\circ) - T = m_1 a \)
2. \( T - m_2 g \sin(25^\circ) = m_2 a \)

### Step 3: Solve for the tension \( T \)

From equation (1):
\[
T = m_1 g \sin(54^\circ) - m_1 a
\]

### Step 4: Substitute \( T \) into equation (2)

Substitute the expression for \( T \) into equation (2):
\[
(m_1 g \sin(54^\circ) - m_1 a) - m_2 g \sin(25^\circ) = m_2 a
\]

### Step 5: Solve for \( m_2 \)

Now, we can solve for \( m_2 \):
\[
m_1 g \sin(54^\circ) - m_1 a = m_2 g \sin(25^\circ) + m_2 a
\]

Let me calculate the value of \( m_2 \).The right-hand mass (\(m_2\)) should be approximately \(2.59 \, \text{kg}\) in order for the system to accelerate downslope at \(0.64 \, \text{m/s}^2\).

Would you like further details or clarification on any of the steps?

Here are some related questions you might find useful:
1. What happens if the angles of the inclines change?
2. How does the system behave if there is friction on the incline?
3. Can the acceleration be negative, and how would that affect the system?
4. How does changing one of the masses affect the acceleration?
5. What if the pulley has mass or friction? How would the equations change?

**Tip:** Always double-check the angle conversion when solving problems involving slopes and inclines in trigonometric functions!

Suppose the angles shown in the figure are 54° and 25°. If the left-hand mass is 1.7 kg, what should the right-hand mass be so that it accelerates downslope at 0.64 m/s²? Express your answer with the appropriate units.

Find the required mass on an inclined plane for a system with angles 54° and 25° to accelerate downslope at 0.64 m/s². Includes step-by-step calculations using Newton's Laws and trigonometric principles.

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