# University of Wisconsin Green Bay

A bullroarer is a device that was used by Native Americans, Aboriginal Australians and others as a tool for communication. It consists of a thin piece of wood tied to the end of a cord, and if it is whirled around in a circle while the wooden piece is also spinning, it makes a loud humming sound. If you make a bullroarer using a 0.065 kg piece of wood, and whirl it over your head in a horizontal circle so that it makes 2.5 revolutions every second, what is the angle that the cord makes with the horizontal? Assume that the radius of the circle is 1.2 m. What is the tension in the cord?

• In this problem, you are asked to relate motion (the bullroarer moves in a circle) to force (tension). Force and motion of a single object are always related through Newton’s Second Law, so this is a force or 2nd Law problem.

If the problem only asked for the angle made by the rope, and therefore did not mention tension or any other force explicitly, after a little bit of thought you should still recognize this as a Second Law problem. Moving in a circle requires an inward force—in this case, tension, which is directed along the same direction as the rope.

As a general rule of thumb, if an object is moving in a uniform (constant speed) circle, forces are most likely to be the interactions that allow you to understand the problem in the most straightforward way.

• Step 1

Your FBD is not yet finished, because tension has both x- and y- components. Continue down to step 2 when you are ready to continue.

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Step 2

In the final FBD drawn here, all forces are divided into components. The contribution each force makes in the x-direction (in the plane of the circle) is shown explicitly, as is the contribution each force makes in the y-direction. The FBD is now a visual representation of ∑F=ma in each direction.

• The key equation for any problem that relates forces and motion is Newton’s Second Law. Regardless of what quantity you are asked to find, begin with the Second Law. If additional information is needed, it will become apparent as you proceed.

• Step 1

At this point, it seems that you have two equations and three unknowns (T, θ, and v). Values for m and r are specified in the problem. But the problem also states that the bull-roarer makes 2.5 revolutions every second, which is an indirect way of specifying velocity. (Click here to see the velocity calculation.) So you actually have two equations in two unknowns. Scroll down to Step 2 to continue the mathematical solution.

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Step 2

At this stage in the problem, we have two unknowns, θ and T, and two unsolved equations:

-Tcosθ = m(-v2/r)
Tsinθ = mg

One approach that always works is to solve one equation for one of the variables and substitute it into the other.

Now that you have solved for one of the unknown variables, substitute it into either of the original equations to solve for the other variable. I will substitute it into the second equation.

T sin(1.90) = mg
T = (0.065 kg)(9.8 m/s2)/ sin(1.90)
T = 20. N

Tension in the cord and angle of the blocks are the only information requested in this problem. No further mathematical solution is necessary.

• As summarized in Newton’s First Law, if there is no net force on an object it will move in a straight line at a constant speed. In this case, although the speed of the wooden piece is constant, its direction is not. Therefore, we know that there is a net force on it. Whenever motion of an object is in a circle, we further know that the net force must be exactly strong enough to provide an inward acceleration of v2/r, where v is the speed of the object around the circle and r is the radius of that circle.

In this problem, it is the horizontal component of tension which provides the inward net force. In addition, the cord must angle slightly below the horizontal in order to also have an upward component to balance the gravitational force.

Mathematically, there were several extra steps in solving this problem. It is frequently true that a problem will describe revolutions, for example, rather than giving speed. As long as you are confident in your basic approach to the problem (Newton’s Second Law) the problem will steer you towards those side steps. In this case, when it came time to fill in numbers, you recognized that revolutions/time is related to speed.