A person pushes a 20.1-kg shopping cart at a constant velocity for a distance of 39.5 m on a flat horizontal surface. She pushes in a direction 27.3 ° below the horizontal. A 40.6-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 find the magnitude of the force that the shopper exerts, we can use the equation for the net force acting on the shopping cart:
Net force = Force exerted by the shopper - Frictional force = 0
Therefore, we can write:
Force exerted by the shopper = Frictional force
So, the magnitude of the force that the shopper exerts is 40.6 N.
To determine the work done by different forces, we can use the equation:
Work = Force x Distance x cos(θ)
where θ is the angle between the force vector and the direction of displacement.
(a) The work done by the pushing force (Force exerted by the shopper) is calculated as follows:
Work by pushing force = Force exerted by the shopper x Distance x cos(θ)
= 40.6 N x 39.5 m x cos(27.3°)
(b) The work done by the frictional force is given by:
Work by frictional force = Frictional force x Distance x cos(180°)
= 40.6 N x 39.5 m x cos(180°)
(c) The work done by the gravitational force is given by:
Work by gravitational force = Force due to gravity x Distance x cos(θ)
= (Mass x Gravity) x Distance x cos(θ)
= (20.1 kg x 9.8 m/s^2) x 39.5 m x cos(θ)
Note that in this case, the gravitational force is acting vertically downward, so the angle θ with respect to the horizontal is 90°.
(d) The work done by the gravitational force is the negative of the work done by the pushing force and the frictional force combined:
Work by gravitational force = -(Work by pushing force + Work by frictional force)
Now, you can calculate the values for (a), (b), (c), and (d) using the given values and formulas.
To answer this question, we need to use the principles of Newton's laws of motion and work done. Let's break it down step by step:
(a) What is the magnitude of the force that the shopper exerts?
To find the magnitude of the force that the shopper exerts, we need to consider the forces acting on the cart. There are two forces involved: the force the shopper exerts and the frictional force opposing the motion.
The horizontal component of the force the shopper exerts can be found using trigonometry. Since the angle is given as 27.3° below the horizontal, we need to find the cosine of this angle:
Cos(27.3°) = Adjacent / Hypotenuse
The adjacent side represents the horizontal component of the force, and the hypotenuse represents the magnitude of the force. Therefore, the horizontal component of the force exerted by the shopper is:
F_horizontal = Magnitude of the force * Cos(27.3°)
To find the magnitude of the force, we can rearrange the formula:
Magnitude of the force = F_horizontal / Cos(27.3°)
(b) What is the work done by the pushing force?
Work done is given by the formula:
Work = Force * Distance * Cos(θ)
Here, the force is the horizontal component of the force exerted by the shopper, the distance is the distance traveled by the shopper, and the angle θ is the angle between the force and the direction of motion (which is 0° in this case since the motion is in the horizontal direction).
Work done by the pushing force = F_horizontal * Distance * Cos(0°)
(c) What is the work done by the frictional force?
The work done by a force can be calculated using the same formula as above:
Work = Force * Distance * Cos(θ)
Here, the force opposing the motion (frictional force) is given as 40.6 N, the distance traveled by the shopper is 39.5 m, and the angle θ between the force of friction and the direction of motion is 180° since friction acts opposite to the direction of motion.
Work done by the frictional force = -40.6 N * 39.5 m * Cos(180°)
The negative sign indicates that the work done by the frictional force is in the opposite direction of motion.
(d) What is the work done by the gravitational force?
The work done by the gravitational force can be calculated using a similar formula:
Work = Force * Distance * Cos(θ)
Here, the force of gravity can be found using the equation:
Force of gravity = Mass * Acceleration due to gravity
Since the cart is on a flat horizontal surface and there is no vertical displacement, the angle θ between the force of gravity and the direction of motion is 90°. Therefore, the work done by the gravitational force is zero.
I hope this helps you understand how to approach this problem and calculate the required quantities.