When pushing a shopping cart, the main force being applied is a non-conservative force. Specifically, it is an applied force from a person pushing the cart that causes it to move. This pushing force is not conservative because energy is constantly being added to the system (the cart) from the person pushing it. Without the input of energy from the pusher, the cart would quickly come to a stop due to frictional forces acting on it.
Why is the force non-conservative?
Forces can be divided into two main categories: conservative forces and non-conservative forces. Conservative forces are forces that conserve energy in a system. This means that the total mechanical energy (kinetic and potential energy) remains constant when only conservative forces act on an object. Examples of conservative forces include gravity, spring forces, and electrostatic forces.
Non-conservative forces, on the other hand, do not conserve mechanical energy. These forces remove or add energy to a system, causing the total mechanical energy to change. Friction, air resistance, and applied forces like pushing are examples of non-conservative forces. When a person pushes a shopping cart, they are exerting an external applied force on the cart. This force does work on the cart to accelerate it forward and give it kinetic energy. But this energy is not conserved – it is dissipated as heat due to friction from the wheels rolling and air resistance slowing the cart down. So the pusher must continuously apply more force to maintain or increase the cart’s speed.
If the force was conservative, the cart would move at a constant speed once the pusher gave it an initial push. But because the pushing force is non-conservative, energy is not conserved and the pusher must keep applying more force to overcome frictional losses and maintain the cart’s motion. This is why the pushing force is considered non-conservative.
Forces acting on a moving shopping cart
When a shopping cart is moving at a constant velocity across a store floor, there are several forces acting on it:
- Pushing force (Fpush) – This is the force exerted by the person pushing the cart forward.
- Friction force (Ffriction) – The friction force acts opposite the direction of motion due to the cart’s wheels rolling on the floor.
- Air resistance (Fair) – The air resistance also acts opposite the direction of motion as the cart moves through the air.
- Normal force (FN) – This is the support force exerted perpendicular to the floor surface. It counteracts the downward force of gravity.
- Gravitational force (Fg) – The force of gravity acts downward through the center of mass of the cart.
These forces can be represented in the following diagram:
| Forces on a Shopping Cart | |
|---|---|
|
According to Newton’s Second Law, the net force on an object is equal to its mass multiplied by its acceleration:
∑F = ma
Where:
- ∑F = Sum of all forces (vector sum)
- m = Mass of the object
- a = Acceleration of the object
For a shopping cart moving at constant velocity, the acceleration is zero. Therefore, the net force is zero:
∑F = 0 = Fpush – Ffriction – Fair + FN – Fg
The pushing force exactly balances out the friction, air resistance, and gravitational forces, resulting in no net acceleration.
Factors affecting the pushing force
Several factors determine how much pushing force needs to be applied to maintain a constant velocity:
- Mass of the cart – Carts with more mass require greater pushing force to overcome friction and air resistance.
- Velocity – Faster moving carts experience larger friction and drag forces, needing more push force.
- Wheel and axle condition – Worn wheels and axles increase friction and rolling resistance, necessitating greater push force.
- Floor surface – Smoother floors like tile or concrete require less force than rougher surfaces like carpet.
- Number of wheels – Carts with more wheels tend to roll more easily, requiring less force.
- Lubrication – Well-lubricated wheel axles and bearings reduce friction, needing less push force.
- Tire pressure – Proper inflation reduces tire deformation, lowering rolling resistance.
By minimizing these sources of friction and resistance, the required pushing force can be reduced. Keeping cart wheels well-maintained and using slippery floor coatings are two strategies stores employ to make carts easier to push with minimal force.
Energy considerations
We can examine the energetics of pushing a shopping cart by considering the work done to move the cart a certain distance. Work is defined as:
W = Fd
Where:
- W = Work done (Joules)
- F = Force applied (Newtons)
- d = Distance moved (meters)
The work done by pushing is transferred to the cart in the form of kinetic energy. The kinetic energy of an object is defined as:
KE = 1/2 mv2
Where:
- KE = Kinetic energy (Joules)
- m = Mass of the object (kg)
- v = Velocity of the object (m/s)
However, the cart’s kinetic energy is constantly being lost due to non-conservative forces like friction heating the wheels and bearings. So the pusher must continuously input work to maintain the cart’s speed.
The power expended by the pusher is the rate at which they do work:
P = W/t
Where:
- P = Power (Watts)
- W = Work done (Joules)
- t = time period over which work is done (seconds)
The power required depends on how quickly the pusher is moving the cart. Quickly accelerating a heavy cart results in a large power expenditure. Pushing at a leisurely, constant speed requires much less power.
Special cases
Pushing a cart up or down a slope
On an inclined surface like a ramp, the force of gravity acting on the cart has a component parallel to the surface. When pushing a cart up a slope, this gravitational force opposes the motion, increasing the required push force. Alternatively, rolling down a slope decreases the pushing force needed, as gravity accelerates the cart.
The parallel gravitational component can be calculated using:
Fg sin(θ)
Where:
- Fg = force of gravity
- θ = angle of incline
This force must be overcome when pushing up an incline but assists motion when going down. The steeper the slope, the larger the effect.
Coasting vs. pushing
If the pusher releases the cart and lets it coast freely, the forces change. There is no longer an applied push force. The cart will gradually slow down as friction and air resistance dissipate its kinetic energy. The deceleration while coasting can be found from:
a = Fnet/m
Where Fnet is the net friction and air resistance. The coasting distance depends on the cart’s initial velocity when released and how quickly these forces decelerate it to a stop.
Coasting requires no energy input from the pusher. But it only lasts until the cart comes to rest. Pushing continuously overcomes dissipative forces, allowing the cart to maintain motion indefinitely (assuming the pusher doesn’t get exhausted).
Turning a cart
When turning a shopping cart, friction and normal forces provide the centripetal force needed to cause circular motion. Friction between the rear wheels and floor keep the back end from sliding out, while friction at the front wheels allows them to pivot. The normal force also increases on the wheels on the inside of the turn to provide the inward centripetal acceleration.
Sharper turns require greater friction and normal forces. This increased resistance results in larger pushing forces needed to maintain velocity during turns. Well-lubricated wheels can allow easier turning with lower forces.
Using physics concepts
The physics involved in pushing shopping carts can be summarized as:
- Newton’s Laws – Pushing force causes cart acceleration. Net force determines change in motion.
- Work, Energy, Power – Pushing does work to give cart kinetic energy. Power needed depends on speed.
- Friction – Friction reduces net force. Minimizing friction decreases required push force.
- Inclines – Gravity assists or hinders motion on slopes. Affects pushing force parallel to surface.
- Centripetal force – Friction and normal force needed for turning carts.
The non-conservative nature of the pushing force means the pusher must continuously input energy to keep the cart moving. This interplay of forces, work, and energy is what determines the effort required to get those groceries home!
Real-world applications
Looking at shopping cart physics helps illuminate principles used in many everyday mechanical systems. Applications include:
- Vehicle propulsion – The forces involved are analogous to accelerating and maintaining speed in cars, bikes, planes, etc.
- Robotics – Mobile robots require constant power input to overcome friction and move across surfaces.
- Conveyor belts – Conveyors use similar principles, often with motors applying the force to move objects.
- Factories – Assembly lines, machine tools, and other systems require power inputs to overcome friction and do work.
- HVAC – Fans and pumps apply forces for moving air and liquids through ductwork and pipes.
- People movement – Escalators, moving walkways, revolving doors also involve continuously applied forces.
So next time you’re pushing a shopping cart, realize you’re demonstrating principles used widely across mechanical engineering and physics!
Conclusion
When pushing a shopping cart at constant velocity, the main force involved is the non-conservative pushing force applied by the person. This force does work on the cart to maintain its motion by overcoming friction and other resistive forces trying to slow it down. The amount of force and power needed depends on factors like the cart’s mass, speed, wheel and axle condition, and slope of the surface. Shopping carts provide an everyday example of basic physics concepts like Newton’s Laws, friction, work, energy, and power in an applied context.