To give a 19-kg child a ride, two teenagers pull on a 3.8-kg sled with ropes, as indicated in the figure . Both teenagers pull with a force of 55 N at an angle of 35 ∘ relative to the forward direction, which is the direction of motion. In addition, the snow exerts a retarding force on the sled that points opposite to the direction of motion, and has a magnitude of 57 N. Find the acceleration of the sled AND child
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To find the acceleration of the sled and child, we need to understand the forces acting on them. Let's break it down step by step.
1. Identify the forces present:
- The force applied by the teenagers pulling the sled (55 N at an angle of 35°).
- The retarding force exerted by the snow (57 N).
- The weight of the child (mg, where m is the mass and g is the acceleration due to gravity).
2. Resolve the force applied by the teenagers:
- Since the force is given at an angle, we need to find its horizontal (Fx) and vertical (Fy) components.
- Fx = F * cos(θ), where F is the force magnitude and θ is the angle.
- Fy = F * sin(θ).
3. Calculate the total vertical force:
- The vertical forces are the weight of the child and the vertical component of the teenagers' force.
- Fy-total = Fy (force exerted by teenagers) - mg (child's weight).
4. Calculate the net horizontal force:
- The horizontal forces are the horizontal component of the teenagers' force and the retarding force exerted by the snow.
- Fx-net = Fx (force exerted by teenagers) - F(retarding force exerted by snow).
5. Apply Newton's second law:
- Calculate the net force in the horizontal direction (Fx-net) and the net force in the vertical direction (Fy-total).
- Use the equation F = ma, where F is the net force and a is the acceleration.
6. Solve for the acceleration:
- Divide the net force in the horizontal direction (Fx-net) by the total mass (mass of child + mass of sled) to find the acceleration.
By following these steps, you should be able to find the acceleration of the sled and child. Make sure to plug in the given values for the mass of the child, mass of the sled, forces applied by the teenagers, and the retarding force exerted by the snow.
To find the acceleration of the sled and child, we will first calculate the net force acting on the sled and child system using Newton's second law of motion.
The net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the sled and child system can be considered as one object.
Net force = mass × acceleration
The net force acting on the sled and child system is the vector sum of the tension forces exerted by the teenagers minus the force of friction due to the snow.
Net force = Tension force - Force of friction
The tension force has both horizontal and vertical components. The vertical component will cancel out because the gravitational force on the child is balanced by the normal force from the ground. Therefore, we only need to consider the horizontal components of the tension forces.
First, let's calculate the horizontal component of the tension forces.
Horizontal component of tension force = Tension force × cos(angle)
Given that the tension force exerted by both teenagers is 55 N at an angle of 35°, the horizontal component of the tension force is:
Horizontal component of tension force = 55 N × cos(35°)
Next, let's calculate the net force.
Net force = (Horizontal component of tension force) - (Force of friction)
= (55 N × cos(35°)) - 57 N
Now, we can use Newton's second law of motion to find the acceleration.
Net force = mass × acceleration
We are given the mass of the sled and child system, which is 19 kg.
(55 N × cos(35°)) - 57 N = 19 kg × acceleration
Finally, we can solve for the acceleration.
acceleration = [(55 N × cos(35°)) - 57 N] / 19 kg
Calculating this value will give you the acceleration of the sled and child system.