A person pushes a 19.5-kg shopping cart at a constant velocity for a distance of 35.6 m on a flat horizontal surface. She pushes in a direction 25.4 ° below the horizontal. A 53.1-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. Fap*cos25.4 - Ff = m*a.
Fap*cos25.4 - 53.1 = 19.5*0 = 0 Fap*cos25.4 = 53.1
Fap = 53.1/cos25.4 = 58.8 N. = Force applied by shopper.
b. Work = Fx * d=58.8*cos25.4*35.6=1891 Joules.
c. Work = Ff*d = 53.1 * 35.6 = 1890 J.
To find the magnitude of the force that the shopper exerts, we need to consider the forces acting on the shopping cart. There are three forces involved: the pushing force, the frictional force, and the gravitational force.
(a) The magnitude of the force that the shopper exerts can be found by using 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. Since the cart is moving at a constant velocity, the acceleration is zero. Therefore, the net force acting on the cart is also zero. This means that the magnitude of the force that the shopper exerts is equal in magnitude but opposite in direction to the sum of the frictional force and the gravitational force.
(b) The work done by the pushing force can be calculated using the formula: Work = Force * Distance * cos(theta), where theta is the angle between the force applied and the direction of motion. In this case, the angle is 25.4° below the horizontal. The distance is given as 35.6 m, and the force applied is the magnitude of the force that the shopper exerts.
(c) The work done by the frictional force can be calculated using the formula: Work = Force * Distance * cos(180°), since the frictional force opposes the motion and acts in the opposite direction. The distance is given as 35.6 m, and the force applied is the magnitude of the frictional force.
(d) The work done by the gravitational force can be calculated using the formula: Work = Force * Distance * cos(θ), where θ is the angle between the force applied and the direction of motion. In this case, the force of gravity acts vertically downward, while the displacement of the shopping cart is horizontally. Since the two directions are perpendicular, the angle between them is 90°. Therefore, the work done by the gravitational force is zero.
To summarize:
(a) The magnitude of the force that the shopper exerts can be found by adding the frictional force and the gravitational force: |Force Shopper| = |Force Friction| + |Force Gravity|.
(b) The work done by the pushing force can be calculated using the formula: Work Pushing = |Force Shopper| * Distance * cos(theta).
(c) The work done by the frictional force can be calculated using the formula: Work Friction = |Force Friction| * Distance * cos(180°).
(d) The work done by the gravitational force is zero.
Now, you can use the given values to calculate each of the above quantities.