Question 7.10: Tipping a vending machine According to the U.S. Consumer Pro......
Tipping a vending machine
According to the U.S. Consumer Product Safety Commission, tipped vending machines caused 37 fatalities between 1978 and 1995 (2.2 deaths per year). Why is tipping vending machines so dangerous? A typical vending machine is 1.83 m high, 0.84 m deep, and 0.94 m wide and has a mass of 374 kg. It is supported by a leg on each of the four corners of its base.
(a) Determine the horizontal pushing force you need to exert on its front surface 1.5 m above the floor in order to just lift its front feet off the surface (so that it will be supported completely by its back two feet).
(b) At what critical angle would it fall forward?
Sketch and translate (a) See the sketch of the situation below. The axis of rotation will go through the back support legs of the vending machine.
Simplify and diagram
Model the vending machine as a rigid body. Three objects exert forces on the vending machine (shown in the side view force diagram below) : the person exerting normal force \vec{N}_{\text {Pon } \mathrm{M}} on the machine, Earth exerting gravitational force \vec{F}_{\text {E on M}} and the floor exerting normal force \vec{N}_{\text {Fon } \mathrm{M}} and static friction force \vec{f}_{\text {s F on } \mathrm{M}} on the machine.
Represent mathematically ( a ) We us e the torque condition of equilibrium to analyze the force needed to tilt the vending machine until the front legs are just barely off the floor. The normal force exerted by the person has a clockwise turning ability while the gravitational force has counterclockwise turning ability. The force exerted by the floor the back legs does not produce a torque, since it is exerted at the axis of rotation.
-N_{\mathrm{P} \text { on } \mathrm{M}} L_{\mathrm{P}} \sin \theta_{\mathrm{P}}+F_{\mathrm{E} \text { on } \mathrm{M}} L_{\mathrm{E}} \sin \theta_{\mathrm{E}}+N_{\text {F on } \mathrm{M}}(0) +f_{\mathrm{s} \mathrm{F} \text { on } \mathrm{M}}(0)=0(b) We can apply the analysis in Testing Experiment Table 7.6 for this situation:
\theta_{\mathrm{C}}=\tan ^{-1}\left(\frac{\text { depth }}{\text { height }}\right)
Table 6.7 Types of collisions.
| Elastic collisions | Inelastic collisions | Totally inelastic collisions |
| Both the momentum and kinetic energy of the system are constant. The internal energy of the system does not change. The colliding objects never stick together.
Examples: There are no perfectly elastic collisions in nature, although collisions between very rigid objects (such as billiard balls) come close. Collisions between atoms or subatomic particles are almost exactly elastic. |
The momentum of the system is constant but the kinetic energy is not. The colliding objects do not stick together. Internal energy increases during the collisions. Examples: A volleyball bouncing off your arms, or you jumping on a trampoline. | These are inelastic collisions in which the colliding objects stick together. Typically, a large fraction of the kinetic energy of the system is converted into internal energy in this type of collision.
Examples: You catching a football, or a car collision where the cars stick together. |
Solve and evaluate (a) Using the torque equation, we can find the normal force that the person needs to exert on the vending machine to just barely lift its front off the floor:
-N_{\mathrm{P} \text { о M }} L_{\mathrm{P}}\left(\frac{1.5 \mathrm{~m}}{L_{\mathrm{P}}}\right)+(3700 \mathrm{~N}) L_{\mathrm{E}}\left(\frac{0.42 \mathrm{~m}}{L_{\mathrm{E}}}\right)
+N_{\mathrm{F} \text { on } \mathrm{M}}(0)+f_{\mathrm{s\text{ }F\text{ }on}\text{ } \mathrm{M}}(0)=0
\text { or } N_{\mathrm{P} \text { on } \mathrm{M}}=1000 \mathrm{~N} \text { or } 220 \mathrm{lb} \text {. }
(b) We find that the critical tipping angle is
\theta_{\text {tipping }}=\tan ^{-1}\left(\frac{\text { depth }}{\text { height }}\right)=\tan ^{-1}\left(\frac{0.42 \mathrm{~m}}{0.92 \mathrm{~m}}\right)=25^{\circ}
Both answers seem reasonable.
Try it yourself: Determine how hard you need to push against the vending machine to keep it tilted at a 25° angle above the horizontal.
Answer: 0 N. A vertical line through the center of mass passes through the axis of rotation, so that the net torque about that axis is zero. However, this equilibrium is unstable and represents a dangerous situation.