Simple Pendulums

Simple Pendulum: Tarzan goes to the pool

Simple pendulums and the radial and tangential axis

The problem

At the indoor pool, there is rope swing hanging from the ceiling that kids use to jump into the pool. They would get up on a high platform, hold onto the rope, step off and swing into the pool. This rope swing is strictly for small kids, but ignoring safety warnings, Tarzan, a 65.0kg adult decides to have a go, holding onto the end of the rope and stepping off the platform. The maximum tension the rope can support is 400N. What is his tangential acceleration just as the rope snaps?

Assume: The rope is has essentially no mass, air drag or frictional forces are negligible, and Tarzan simple steps off the platform and does not launch or propel himself off.

Step 1: Draw the Diagram

by Victoria (Hua) Chen

SN: 31688147

Physics 101 L2D

LO 2

Draw the diagram including the rope, the platform and Tarzan and then label the appropriate forces

Notice that tension and mg are not equal

Step 2: Draw the radial axis and tangential axis

We must then label the two components of the weight, the radial (blue) axis, which runs in the along the rope, and the tangential axis (green), which runs perpendicular to the radial axis

Since the radial axis forms the angle theta with the vertical mg, using our trigonometry rules of

cos(theta)=adjacent/hypotenuse and

sin(theta)=opposite/hypotenuse

we can view the radial axis is the "adjacent", the tangential axis as the "opposite" and the vertical component mg as the "hypotenuse so that the

radial axis = (mg)cos(theta)

tangential axis= (mg)sin(theta)

Solving the equation

Conclusion

Step 4: Finding acceleration

The focus of this LO was to fully understand the components of the simple pendulum, and in particular the radial and tangential axis. It is important to understand that the weight mg is split up into these two separate components. By illustrating that the radial weight component is equal to our tension force and that the tangential weight component gives us our restoring force, which allows us to solve for our tangential acceleration, we are can make use of the only given information to solve this problem.

The tangential component of the weight acts as the restoring force to pull the mass, Tarzan, back to equilibrium.

Newton's second law states

Fnet= -(mg)sin(theta)=ma

*note the (-), which is important since this is a restoring force

the mass cancels leaving us with a formula for tangential acceleration

a= -gsin(theta)

Now we can plug in our value for theta calculated in step 3 and solve to find our answer:

a= - 7.63 m/s^2

Step 3: Equate radial axis weight to tension

Since there is no motion on the radial axis, the tension and the weight along the radial axis must be equal, as they point in opposite directions

We can then equate the two components, plug in our values and solve for theta. We want to find the tangential acceleration as the rope snaps, so our Tmax value must be 400N.