A person pushes a 13.7-kg shopping cart at a constant velocity for a distance of 16.3 m on a flat horizontal surface. She pushes in a direction 22.2 ° below the horizontal. A 48.7-N frictional force opposes the motion of the cart. (a) What is the magnitude of the force that the shopper exerts? Determine the work done by (b) the pushing force, (c) the frictional force, and (d) the gravitational force.
To find the magnitude of the force that the shopper exerts, we can make use of Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the shopping cart is moving at a constant velocity, which means its acceleration is zero. Therefore, the net force acting on the cart must also be zero. The forces acting on the cart are the force applied by the shopper (in the direction of motion) and the frictional force opposing the motion.
We can resolve these forces into their horizontal and vertical components:
1. The force applied by the shopper has a horizontal component Fx and a vertical component Fy. The horizontal component Fx is equal to the magnitude of the applied force multiplied by the cosine of the angle below the horizontal: Fx = F * cos(22.2°).
2. The frictional force opposes the motion and is directed opposite to the applied force. Therefore, its horizontal component is also in the opposite direction and has a magnitude equal to the magnitude of the frictional force: Ffriction_x = -48.7 N.
Since the net force is zero, the horizontal components of the applied force and the frictional force must be equal: Fx - Ffriction_x = 0.
Solving for Fx, we have:
Fx - 48.7 N = 0,
Fx = 48.7 N.
Therefore, the magnitude of the force that the shopper exerts is 48.7 N.
Now, let's determine the work done by each of the forces.
(b) The work done by the pushing force is given by the equation:
Work_pushing = Force_pushing * displacement * cos(θ),
where θ is the angle between the force and the displacement. In this case, the angle is 0° since the force is in the same direction as the displacement.
Therefore, Work_pushing = 48.7 N * 16.3 m * cos(0°) = 794.81 J.
(c) The work done by the frictional force can be calculated using:
Work_friction = Force_friction * displacement * cos(180°),
where θ is the angle between the force and the displacement. Since the frictional force is in the opposite direction of motion, the angle is 180°.
Therefore, Work_friction = -48.7 N * 16.3 m * cos(180°) = -794.81 J.
(d) The work done by the gravitational force is given by:
Work_gravity = Force_gravity * displacement * cos(θ),
where θ is the angle between the force of gravity and the displacement. In this case, the displacement is horizontal, and the angle between the force of gravity and the displacement is 90°.
Therefore, Work_gravity = 0 N * 16.3 m * cos(90°) = 0 J.
Hence, the work done by the pushing force is 794.81 J, the work done by the frictional force is -794.81 J, and the work done by the gravitational force is 0 J.
Let's break down the problem into smaller steps to find the answers:
Step 1: Calculate the magnitude of the force that the shopper exerts.
The forces acting on the shopping cart are the force exerted by the shopper and the frictional force. Since the cart is moving at a constant velocity, the net force is zero.
Therefore, the magnitude of the force that the shopper exerts is equal to the magnitude of the frictional force:
Magnitude of force exerted by shopper = Magnitude of frictional force = 48.7 N
Step 2: Determine the work done by the pushing force.
The work done by a force is given by the formula: work = force * distance * cos(theta), where theta is the angle between the force and displacement.
In this case, the angle is 22.2° below the horizontal, so the angle between the force and the displacement is 90° - 22.2° = 67.8°.
Work done by the pushing force = force exerted by the shopper * distance * cos(theta)
Work done by the pushing force = 48.7 N * 16.3 m * cos(67.8°)
Step 3: Determine the work done by the frictional force.
Since the cart is moving at a constant velocity, the work done by the frictional force is equal to the negative of the work done by the pushing force.
Work done by the frictional force = - Work done by the pushing force
Step 4: Determine the work done by the gravitational force.
The gravitational force is acting vertically downward and the cart is moving horizontally. Therefore, the angle between the gravitational force and the displacement is 90°.
The work done by a force acting at an angle of 90° is zero.
Work done by the gravitational force = 0
To summarize:
(a) Magnitude of the force that the shopper exerts = 48.7 N
(b) Work done by the pushing force = 48.7 N * 16.3 m * cos(67.8°)
(c) Work done by the frictional force = - Work done by the pushing force
(d) Work done by the gravitational force = 0