The Physics of a Dog Tug of War

 My family recently got a new dog which means that we now have two little dogs - Bonnie and Maisy. They love playing with each other, and their favorite toy to ‘share’ is a frog-shaped rope (see photos below). Although they’re approximately the same size, the older dog, Maisy, always seems to win the tug of war. I got to thinking about how physics (specifically the principle of torque) may play a role in her domination of the toy war. Torque is a good way to convey the force that Maisy exerts on the rope as she shakes her head back and forth in a somewhat circular motion. In this model, the puppy’s mouth grabbing one end of the rope is the pivot point and the force is being applied by Maisy on the other end of the rope Maisy has found that by grabbing onto the very end of the rope, rather than the middle of the rope, she can more forcefully tug the rope and successfully yank it from Bonnie's mouth. This relationship can be explained by the principle of torque, and the relationship between the lever arm radius and the corresponding amount of torque produced.

Measurements

• From knot to knot (the points at which the dogs grab), the rope is 0.25 m long

Assumptions

• Maisy is applying a force perpendicular to the lever arm

• Bonnie is grabbing onto the other end of the rope at a constant point (the pivot point)

• We do not know the force (F) exerted by Maisy, so we are using F in place of the actual value

• The force (F) exerted by Maisy is the same in both scenarios

Calculations

The torque of Scenario 1: 

In scenario 1, Maisy would grab onto the rope at a point closer to the pivot point, thus making the lever arm (r) shorter.

Torque = rFsin

Torque = 0.10F

The torque of Scenario 2: 

In scenario 2, Maisy would grab onto the rope at the point farthest from the pivot point, thus making the lever arm (r) longer.

Torque = rFsin

Torque = 0.25F

Thus, the torque in Scenario 2 is 2.5 times that of Scenario 1.

Conclusion

Despite not knowing the exact force Maisy is exerting on the rope as she pulls, we can use the equation for torque to look at the relationship between where she grabs the rope and the torque exerted. There is a direct relationship between the length of the lever arm and the torque, suggesting that Maisy exerts a greater torque (rotational force) by grabbing onto the end of the rope as opposed to a location closer to the point of rotation (i.e. where Bonnie is holding onto the rope). Thus, in an attempt to win the tug of war, dogs unknowingly use the laws of physics and elongate their lever arm to win!