To find the tension on the rope, we need to first calculate the force of kinetic friction acting on the sledge. We can use the formula for kinetic friction:
\( f_k = \mu_k \cdot N \)
where \( f_k \) is the force of kinetic friction, \( \mu_k \) is the coefficient of kinetic friction, and \( N \) is the normal force.
The normal force is the force exerted by the surface perpendicular to it. In this case, the sledge is on a horizontal surface, so the normal force is equal in magnitude and opposite in direction to the vertical force due to gravity:
\( N = mg \)
where \( m \) is the mass of the sledge and \( g \) is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values, we have:
\( N = 18.9 \, \text{kg} \times 9.8 \, \text{m/s}^2 \)
Next, we can calculate the force of kinetic friction:
\( f_k = 0.500 \times 18.9 \, \text{kg} \times 9.8 \, \text{m/s}^2 \)
Now, we need to find the force along the incline (parallel to the inclined plane). This force is equal in magnitude to the tension in the rope:
\( T = f_k + mg \sin(\theta) \)
where \( T \) is the tension, \( \theta \) is the angle of inclination (19.3°), and \( \sin(\theta) \) represents the component of gravity along the incline.
Finally, we can calculate the tension on the rope:
\( T = 0.500 \times 18.9 \, \text{kg} \times 9.8 \, \text{m/s}^2 + 18.9 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times \sin(19.3°) \)
By evaluating this expression, we can find the numerical value of the tension on the rope.