A person pushes a 24.7-kg shopping cart at a constant velocity for a distance of 34.0 m on a flat horizontal surface. She pushes in a direction 23.2 ° below the horizontal. A 48.5-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.
a. F*cos23.2-48.5 = m*a = m*0 = 0
Fcos23.2 = 48.5
F = 48.5/cos23.2 = 52.8 N.
b. Work = F*d = 52.8 * 34 = 1795 Joules.
c. Work = Fk * d = 48.5 * 34 = 1649 J.
To find the magnitude of the force that the shopper exerts, we need to consider the forces acting on the shopping cart.
Let's break down the forces:
1. The pushing force: This is the force exerted by the shopper. We know that the force has a magnitude equal to the force of friction in order to maintain a constant velocity. Since the frictional force opposes the motion and has a magnitude of 48.5 N, the magnitude of the pushing force is also 48.5 N.
2. The gravitational force: This force is acting vertically downward. The force of gravity can be calculated using the formula F = m * g, where m is the mass of the shopping cart and g is the acceleration due to gravity (approximately 9.8 m/s²). Therefore, the magnitude of the gravitational force is F_grav = 24.7 kg * 9.8 m/s².
3. The frictional force: The frictional force opposes the motion of the cart and has a magnitude of 48.5 N.
Now let's calculate the work done by each force:
(a) The work done by the pushing force: The work done by a constant force is given by the formula W = F * d * cos(θ), where F is the magnitude of the force, d is the distance, and θ is the angle between the force and the displacement. In this case, θ is the angle below the horizontal, so we can use the angle given in the problem, which is 23.2°. Therefore, the work done by the pushing force is W_push = 48.5 N * 34.0 m * cos(23.2°).
(b) The work done by the frictional force: Since the frictional force opposes the motion of the cart, the angle between the frictional force and the displacement is 180°. Therefore, the work done by the frictional force is W_friction = -48.5 N * 34.0 m * cos(180°).
(c) The work done by the gravitational force: Since the displacement is horizontal and the gravitational force is acting vertically downward, the angle between them is 90°. Therefore, the work done by the gravitational force is W_grav = 0, as there is no displacement in the direction of the gravitational force.
Now you can use these formulas and calculations to find the values for (b), (c), and (d).