A dray horse is being pulled by a rope across a level plow field by a force of 750.0N exerted at an angle of 47° above the horizontal. If the horse’s velocity is constant and the coefficient of friction is ì=.21, determine the mass of the horse.
To determine the mass of the horse, we need to analyze the forces acting on it and use Newton's second law of motion. Here's how we can solve the problem step by step:
Step 1: Identify the forces acting on the horse:
- The applied force pulling the horse (750.0 N) at an angle of 47° above the horizontal.
- The gravitational force pulling the horse downwards (weight).
Step 2: Calculate the gravitational force (weight) acting on the horse:
- The weight of the horse is given by the formula: weight = mass x acceleration due to gravity.
- In this case, the acceleration due to gravity is approximately 9.8 m/s^2.
- As the horse is on a level field, the weight is balanced by the normal force exerted by the ground, so they cancel each other out.
Step 3: Determine the net force acting on the horse:
- The net force is the vector sum of all the forces acting on the horse.
- In the horizontal direction, the applied force and the force of friction are the only forces, so the net force is given by: net force = applied force - force of friction.
- In the vertical direction, since the horse's velocity is constant, the net force is zero, therefore: net force = applied force in the vertical direction - weight.
Step 4: Calculate the force of friction:
- The force of friction is given by the formula: force of friction = coefficient of friction x normal force.
- The normal force is equal to the weight, so: force of friction = coefficient of friction x weight.
Step 5: Separate the applied force into horizontal and vertical components:
- The applied force can be resolved into horizontal and vertical components using trigonometry.
- The horizontal component is given by: applied force (horizontal) = applied force x cos(angle).
- The vertical component is given by: applied force (vertical) = applied force x sin(angle).
Step 6: Write the equations for the net force in the horizontal and vertical directions:
- In the horizontal direction: net force (horizontal) = applied force (horizontal) - force of friction = 0.
- In the vertical direction: net force (vertical) = applied force (vertical) - weight = 0.
Step 7: Solve the equations:
- Set up the equation for net force (horizontal): applied force (horizontal) - force of friction = 0.
- Since the applied force (horizontal) is equal to the horizontal component of the applied force, and force of friction is given by the formula, substitute the appropriate values.
- Solve the equation for the force of friction.
- Use the equation for force of friction to solve for the weight of the horse.
Step 8: Calculate the mass of the horse:
- Use the equation weight = mass x acceleration due to gravity to determine the mass of the horse.
- Rearrange the equation to solve for mass.
By following these steps and plugging in the appropriate values, you'll be able to determine the mass of the horse.