A person pushes a 16.0-kg shopping cart at a constant velocity for a distance of 18.0 m. She pushes in a direction 20.0° below the horizontal. A 40.0-N frictional force opposes the motion of the cart.
(a) What is the magnitude of the force that the shopper exerts?
(b) Determine the work done by the pushing force.
(c) Determine the work done by the frictional force.
d) Determine the work done by the gravitational force.
horizontal force=40N
pushing force=40/cos20
work done=40*18 Joules
work done by friction=40*18
no work done by gravity.
To solve this problem, we need to use the concept of work and the equations related to it. Work is defined as the product of the force applied on an object and the displacement of the object.
(a) To find the magnitude of the force that the shopper exerts, we need to consider the forces acting on the shopping cart. The shopper exerts a force to push the cart, and there is a frictional force opposing the motion.
First, let's consider the forces in the horizontal direction. The horizontal component of the force that the shopper exerts cancels out the horizontal component of the frictional force. The horizontal component of the frictional force can be found using trigonometry:
F_friction_horizontal = F_friction * cos(20°)
Substituting the given values, we get:
F_friction_horizontal = 40 N * cos(20°)
Next, the shopper's force in the horizontal direction must balance the horizontal component of the frictional force:
Force_shopper_horizontal = F_friction_horizontal
Therefore,
Force_shopper_horizontal = 40 N * cos(20°)
Now, to find the magnitude of the force that the shopper exerts, we need to find the total force exerted by the shopper. Since the force and displacement are in the same direction (horizontal), the magnitude of the force is equal to the magnitude of the horizontal component of the force:
Magnitude of the force that the shopper exerts = Force_shopper_horizontal
(b) The work done by the pushing force can be calculated using the equation:
Work = Force * Displacement * cos(θ)
Where:
Force = Magnitude of the force that the shopper exerts
Displacement = Distance traveled by the cart (18.0 m)
θ = Angle between the force and the displacement (0°, as the force and displacement are in the same direction)
Therefore,
Work = Magnitude of the force that the shopper exerts * Displacement * cos(0°)
(c) The work done by the frictional force can be calculated using the same equation:
Work = Force * Displacement * cos(θ)
Where:
Force = Magnitude of the frictional force (40 N)
Displacement = Distance traveled by the cart (18.0 m)
θ = Angle between the force and the displacement (180°, as the force and displacement are in opposite directions)
Therefore,
Work = Magnitude of the frictional force * Displacement * cos(180°)
(d) The work done by the gravitational force is given by the equation:
Work = Force * Displacement * cos(θ)
Where:
Force = Weight of the cart (mg, where m is the mass and g is the acceleration due to gravity)
Displacement = Distance traveled by the cart (18.0 m)
θ = Angle between the force and the displacement (90°, as the force and displacement are perpendicular)
Therefore,
Work = Weight of the cart * Displacement * cos(90°)
Note: The weight of the cart can be calculated using the equation:
Weight = mass * gravity acceleration
Substituting the given values, we can find the work done by the gravitational force in this case.