A person pushes a 23.5-kg shopping cart at a constant velocity for a distance of 30.3 m on a flat horizontal surface. She pushes in a direction 31.5 ° below the horizontal. A 37.3-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 determine the magnitude of the force that the shopper exerts, we need to analyze the forces acting on the shopping cart.
a) The forces acting on the shopping cart are the pushing force (F_push), the frictional force (F_friction), and the gravitational force (F_gravity).
The gravitational force acting on the cart can be calculated using the formula: F_gravity = m * g, where m is the mass of the cart and g is the acceleration due to gravity (9.8 m/s²). Since the cart is on a flat horizontal surface, the vertical component of the gravitational force cancels out with the normal force from the surface, leaving only the horizontal component.
F_gravity = 23.5 kg * 9.8 m/s² = 230.3 N
The frictional force is given as 37.3 N opposing the motion.
F_friction = 37.3 N
To determine the pushing force, we need to resolve it into its horizontal and vertical components. The horizontal component is given by F_push_horizontal = F_push * cos(31.5°), where F_push is the magnitude of the pushing force.
F_push_horizontal = F_push * cos(31.5°)
Since the cart is moving at a constant velocity, the net force acting on it is zero. So, the horizontal component of the pushing force is equal to the frictional force.
F_push_horizontal = F_friction
Therefore, F_push * cos(31.5°) = 37.3 N
Now, we can solve for the magnitude of the pushing force, F_push.
F_push = 37.3 N / cos(31.5°)
b) The work done by the pushing force can be calculated using the formula: work = force * distance * cos(angle), where the angle is the angle between the force and the direction of motion.
Work_push = F_push * distance * cos(0°)
Since the cart is pushed in a direction 31.5° below the horizontal, the angle between the force and the direction of motion is 0°. Therefore, cos(0°) = 1.
Work_push = F_push * distance
c) The work done by the frictional force is equal to the negative of the product of the frictional force and the distance.
Work_friction = -F_friction * distance
d) The work done by the gravitational force is equal to the product of the gravitational force component in the direction of motion and the distance.
Work_gravity = F_gravity * distance * cos(angle)
Since the gravitational force is acting vertically and the motion is horizontal, the angle between the force and the direction of motion is 90°. Therefore, cos(90°) = 0.
Work_gravity = 0
To calculate the actual work done by the gravitational force, you should consider any vertical displacement of the cart. However, since the cart is moving on a flat horizontal surface, there is no vertical displacement, and thus, no work is done by the gravitational force in this case.
To summarize:
a) The magnitude of the force that the shopper exerts is F_push = 37.3 N / cos(31.5°).
b) The work done by the pushing force is Work_push = F_push * distance.
c) The work done by the frictional force is Work_friction = -F_friction * distance.
d) The work done by the gravitational force is Work_gravity = 0.
To solve this problem, we need to break it down into several parts.
(a) Magnitude of the force that the shopper exerts:
Since the cart is moving at a constant velocity, we know that the net force acting on it is zero. The pushing force and the frictional force are the only horizontal forces acting on the cart. Thus, the magnitude of the force that the shopper exerts is equal to the magnitude of the frictional force. So, the magnitude of the force that the shopper exerts is 37.3 N.
To determine the work done by each force, we need to use the formula:
Work = Force * Distance * Cosine(theta)
where:
- Work is the energy transferred by the force.
- Force is the magnitude of the force.
- Distance is the distance over which the force is applied.
- Theta is the angle between the force and the displacement.
(b) Work done by the pushing force:
The pushing force is applied in the direction 31.5° below the horizontal. Since the cart moves in the horizontal direction, the angle between the force and the displacement is 0°. Plugging in the values:
Work pushing force = 37.3 N * 30.3 m * Cos(0°)
= 37.3 N * 30.3 m * 1
= 1131.9 J
The work done by the pushing force is 1131.9 Joules.
(c) Work done by the frictional force:
Similar to the pushing force, the frictional force is also acting horizontally. So, the angle between the force and the displacement is 0°.
Work frictional force = 37.3 N * 30.3 m * Cos(0°)
= 37.3 N * 30.3 m * 1
= 1131.9 J
The work done by the frictional force is also 1131.9 Joules.
(d) Work done by the gravitational force:
Since the cart is moving horizontally, the gravitational force does not contribute to the work done.
Therefore, the work done by the gravitational force is zero.
To summarize:
(a) The magnitude of the force that the shopper exerts is 37.3 N.
(b) The work done by the pushing force is 1131.9 Joules.
(c) The work done by the frictional force is 1131.9 Joules.
(d) The work done by the gravitational force is zero.