A tin of antioxidants (m1 " 1.0 kg) on a fric- tionless inclined surface is con- nected to a tin of corned beef (m2  2.0 kg). The pulley is massless and frictionless. An upward force of magnitude F " 6.0 N acts on the  corned beef tin, which has a down- ward acceleration of 5.5 m/s2  . What are (a) the tension in the connecting cord and (b) angle b?( degree from )

To solve this problem, we need to analyze the forces acting on each tin and apply Newton's second law to both. Let's denote the following:

- \( m_1 = 1.0 \, \text{kg} \) (mass of the tin of antioxidants)
- \( m_2 = 2.0 \, \text{kg} \) (mass of the tin of corned beef)
- \( F = 6.0 \, \text{N} \) (upward force acting on the tin of corned beef)
- \( a = 5.5 \, \text{m/s}^2 \) (downward acceleration of the tin of corned beef)
- \( T \) (tension in the connecting cord)
- \( \theta \) (angle of the incline)

### Part (a): Finding the Tension in the Connecting Cord

1. **For the tin of corned beef (m2):**

Let's analyze the forces acting on the tin of corned beef (\( m_2 \)):

- The upward force \( F = 6.0 \, \text{N} \)
- The tension \( T \) in the cord acting upwards
- The gravitational force \( m_2 g \) acting downwards, where \( g = 9.8 \, \text{m/s}^2 \)

Applying Newton's second law (\( \Sigma F = m a \)):

\[ F + T - m_2 g = -m_2 a \]

Here, the acceleration \( a \) is downward, so we take it as negative:

\[ 6.0 + T - 2.0 \times 9.8 = -2.0 \times 5.5 \]

\[ 6.0 + T - 19.6 = -11 \]

\[ T - 13.6 = -11 \]

\[ T = 2.6 \, \text{N} \]

So, the tension in the cord is \( T = 2.6 \, \text{N} \).

### Part (b): Finding the Angle of the Incline \( \theta \)

2. **For the tin of antioxidants (m1):**

Let's analyze the forces acting on the tin of antioxidants (\( m_1 \)):

- The tension \( T = 2.6 \, \text{N} \) acting up the incline
- The component of gravitational force \( m_1 g \sin \theta \) acting down the incline
- The normal force and the component of gravitational force perpendicular to the incline (not affecting the motion along the incline)

The acceleration \( a \) of this tin will be the same as that of the corned beef tin but directed up the incline:

Applying Newton's second law along the incline for \( m_1 \):

\[ T - m_1 g \sin \theta = m_1 a \]

Substitute the known values:

\[ 2.6 - 1.0 \times 9.8 \sin \theta = 1.0 \times 5.5 \]

\[ 2.6 - 9.8 \sin \theta = 5.5 \]

\[ -9.8 \sin \theta = 5.5 - 2.6 \]

\[ -9.8 \sin \theta = 2.9 \]

\[ \sin \theta = -\frac{2.9}{9.8} \]

\[ \sin \theta = -0.2959 \]

Since the sine of an angle is negative in the third and fourth quadrants, and considering the physical context where angles in inclined planes are positive, we take the positive equivalent of the angle (since we're dealing with the absolute incline angle):

\[ \theta = \arcsin(0.2959) \]

\[ \theta \approx 17.23^\circ \]

So, the angle of the incline is approximately \( 17.23^\circ \).

### Summary:
(a) The tension in the connecting cord is \( 2.6 \, \text{N} \).

(b) The angle of the incline \( \theta \) is approximately \( 17.23^\circ \).

Would you like more details or have any questions?

### Relative Questions:
1. How do you apply Newton's second law to inclined planes?
2. Why do we consider the acceleration direction when applying Newton's second law?
3. What is the role of tension in a pulley system with an inclined plane?
4. How does the angle of incline affect the tension in the cord?
5. Can you explain the significance of the components of gravitational force on an inclined plane?
6. How do we determine the direction of forces when dealing with inclined planes?
7. Why do we take the sine component of gravitational force on an inclined plane?
8. How does friction affect the motion on an inclined plane, and what would change in this problem if friction were considered?

### Tip:
When dealing with problems involving inclined planes, always break down the forces into components parallel and perpendicular to the incline. This simplifies the application of Newton's laws.

Learn how to solve a physics problem involving forces and motion on an inclined plane. Calculate the tension in a cord and determine the angle of the incline using Newton's second law. Suitable for high school students studying mechanics.

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