To solve this problem, we need to find the minimum angle $\theta$ at which the child needs to pull the sled to overcome static friction. We are given the following:

• Mass of the sled, $m = 1.65 \, \text{kg}$
• Tension in the rope, $T = 12.3 \, \text{N}$
• Coefficient of static friction, $\mu_s = 1$

Step 1: Forces Acting on the Sled

When the child pulls the sled, two major forces affect its movement:

1. Gravitational Force: The weight of the sled is $W = mg$, where $g = 9.8 \, \text{m/s}^2$.

   So, $W = 1.65 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 16.17 \, \text{N}$

2. Static Friction Force: The maximum static friction force is $F_{\text{friction}} = \mu_s N$, where $N$ is the normal force.

Step 2: Components of the Tension Force

The tension force $T$ is applied at an angle $\theta$, so it has two components:

• Horizontal Component: $T_x = T \cos \theta$
• Vertical Component: $T_y = T \sin \theta$

Step 3: Normal Force

The normal force $N$ is the force exerted by the ground on the sled. It's affected by both the weight of the sled and the vertical component of the tension force: $N = W - T_y = W - T \sin \theta$ Substitute the values: $N = 16.17 \, \text{N} - 12.3 \sin \theta$

Step 4: Condition for Overcoming Static Friction

For the sled to start moving, the horizontal component of the tension force $T_x$ must be equal to the maximum static friction force $F_{\text{friction}}$: $T_x = F_{\text{friction}} = \mu_s N$ Substitute $T_x = T \cos \theta$ and $F_{\text{friction}} = \mu_s N$: $T \cos \theta = \mu_s (W - T \sin \theta)$ Since $\mu_s = 1$, this simplifies to: $T \cos \theta = W - T \sin \theta$

Step 5: Solve for $\sin \theta$

Now we solve the equation $T \cos \theta = W - T \sin \theta$. First, rearrange it as: $T \cos \theta + T \sin \theta = W$ Factor out $T$: $T (\cos \theta + \sin \theta) = W$ Solve for $\cos \theta + \sin \theta$: $\cos \theta + \sin \theta = \frac{W}{T} = \frac{16.17}{12.3} \approx 1.315$ Now, we use the identity $\sin \theta + \cos \theta = \sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right)$ to relate this expression, but since the angle requires finding components, we will work to numerically simplify.