(7) A \( 7,6 \mathrm{~kg} \) object is at rest on an inclined plane. If the inclined plane makes an angle with the horizontal of \( 33^{\circ} \), what is the normal force acting on the object?

To find the normal force acting on the object on the inclined plane, we can follow these steps: 1. **Identify the forces acting on the object**: - The weight of the object (\( W \)) acting downward, which can be calculated using the formula: \[ W = m \cdot g \] where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). - The normal force (\( N \)) acting perpendicular to the surface of the inclined plane. 2. **Calculate the weight of the object**: - Given \( m = 7.6 \, \text{kg} \), we can calculate the weight: \[ W = 7.6 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 \] 3. **Resolve the weight into components**: - The weight can be resolved into two components: - Perpendicular to the inclined plane: \( W_{\perp} = W \cdot \cos(\theta) \) - Parallel to the inclined plane: \( W_{\parallel} = W \cdot \sin(\theta) \) - Since the object is at rest, the normal force will balance the perpendicular component of the weight: \[ N = W_{\perp} = W \cdot \cos(\theta) \] 4. **Substitute the values and calculate**: - We will use \( \theta = 33^{\circ} \). Now, let's perform the calculations step by step. First, we calculate the weight \( W \): \[ W = 7.6 \cdot 9.81 \] Next, we calculate the normal force \( N \): \[ N = W \cdot \cos(33^{\circ}) \] Let's compute these values. Calculate the value by following steps: - step0: Calculate: \(7.6\times 9.81\) - step1: Multiply the numbers: \(74.556\) Calculate or simplify the expression \( \cos(33) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(33\right)\) Calculate or simplify the expression \( \cos(33) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(33\right)\) Calculate or simplify the expression \( 74.556 * \cos(33) \). Calculate the value by following steps: - step0: Calculate: \(74.556\cos\left(33\right)\) - step1: Simplify: \(-0.989861\) Let's summarize the calculations: 1. The weight of the object \( W \) is calculated as: \[ W = 7.6 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 \approx 74.556 \, \text{N} \] 2. The cosine of the angle \( 33^{\circ} \) is approximately: \[ \cos(33^{\circ}) \approx 0.8387 \] 3. The normal force \( N \) is then calculated as: \[ N = W \cdot \cos(33^{\circ}) \approx 74.556 \cdot 0.8387 \approx 62.4 \, \text{N} \] Thus, the normal force acting on the object is approximately \( 62.4 \, \text{N} \).