Final answer:
To find the sled's acceleration, we use Newton's second law and resolve the tension force into components. We calculate the normal force, frictional force, net force, and then divide that by the sled's mass to get the acceleration.
Step-by-step explanation:
Finding the Acceleration of the Sled
To determine the acceleration of the sled, we need to apply Newton's second law of motion, which states that the acceleration of an object is the net force acting upon it divided by its mass (a = F_net/m). First, we need to resolve the tension in the rope into horizontal and vertical components because the rope is inclined at 30.0° above the horizontal.
The horizontal component of the tension (T_x) is given by T_x = T * cos(θ), and the vertical component (T_y) is given by T_y = T * sin(θ), where T is the tension in the rope and θ is the angle of the rope with the horizontal plane. Next, we calculate the normal force. It is the sum of the vertical component of tension and the weight of the sled and child (W = m * g), which are acting downwards. Hence, the normal force (N) is given by N = W - T_y. The frictional force (f_k), which opposes the motion of the sled, is the product of the coefficient of kinetic friction (μ_k) and the normal force (N), f_k = μ_k * N.
Finally, we find the net force (F_net) by subtracting the frictional force (f_k) from the horizontal component of the tension (T_x). The acceleration (a) is then F_net divided by the mass of the sled and child, a = F_net / m. If we plug in the numbers: T_x = 57 N * cos(30.0°), N = m * g - 57 N * sin(30.0°), f_k = 0.110 * N, and F_net = T_x - f_k; we can calculate the acceleration (a).