A person pushes a 23.4-kg shopping cart at a constant velocity for a distance of 31.1 m on a flat horizontal surface. She pushes in a direction 34.8 ° below the horizontal. A 38.5-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 solve this problem, we need to break it down into different components:

(a) To find the magnitude of the force that the shopper exerts, we need to consider the forces acting in the horizontal direction. The gravitational force acting on the cart has two components: one in the horizontal direction and one in the vertical direction. The horizontal component of the gravitational force is equal and opposite to the force that the shopper exerts, which allows the cart to move at a constant velocity.

First, we need to find the horizontal component of the gravitational force:
F_horizontal = m * g * sin(θ),
where m is the mass of the cart (23.4 kg), g is the acceleration due to gravity (9.8 m/s²), and θ is the angle below the horizontal (34.8°).

F_horizontal = 23.4 kg * 9.8 m/s² * sin(34.8°)

Next, since the cart is moving at a constant velocity, the net force in the horizontal direction is zero. Therefore, the magnitude of the force that the shopper exerts is equal to the magnitude of the frictional force (opposing the motion of the cart), which is given as 38.5 N.

So, the magnitude of the force that the shopper exerts is 38.5 N.

(b) To determine the work done by the pushing force, we use the formula:
Work = force * distance * cos(θ),
where force is the magnitude of the pushing force (38.5 N), distance is the distance traveled by the cart (31.1 m), and θ is the angle between the force and the direction of motion (0°, since the force and motion are in the same direction).

Work = 38.5 N * 31.1 m * cos(0°)

(c) The work done by the frictional force is given by:
Work = force * distance * cos(180°),
since the frictional force is acting in the opposite direction of the motion.

Work = 38.5 N * 31.1 m * cos(180°)

(d) The work done by the gravitational force is calculated by the formula:
Work = force * distance * cos(θ),
where force is the magnitude of the gravitational force in the horizontal direction (F_horizontal), distance is the distance traveled by the cart (31.1 m), and θ is the angle between the force and the direction of motion (180°, since the force and motion are in the opposite directions).

Work = F_horizontal * 31.1 m * cos(180°)

Solving these equations will give us the values for (b), (c), and (d).