THE BATON TWIRLER GOAL Calculate a moment of inertia. PROBLEM In an effort to be the star of the halftime show, a majorette twirls an unusual baton made up of four balls fastened to the ends of very light rods (Fig. 8.24). Each rod is 1.0 m long. (a) Find the moment of inertia of the baton about an

Question

THE BATON TWIRLER

GOAL Calculate a moment of inertia.

PROBLEM In an effort to be the star of the halftime show, a majorette twirls an unusual baton made up of four balls fastened to the ends of very light rods (Fig. 8.24). Each rod is [latex]1.0 \mathrm{~m}[/latex] long. (a) Find the moment of inertia of the baton about an axis perpendicular to the page and passing through the point where the rods cross. (b) The majorette tries spinning her strange baton about the axis [latex]O O^{\prime}[/latex], as shown in Figure [latex]8.25[/latex] on page 242 . Calculate the moment of inertia of the baton about this axis.

STRATEGY In Figure [latex]8.24[/latex], all four balls contribute to the moment of inertia, whereas in Figure [latex]8.25[/latex], with the new axis, only the two balls on the left and right contribute. Technically, the balls on the top and bottom in Figure [latex]8.25[/latex] still make a small contribution because they're not really point particles. However, their contributions can be neglected because the distance from the axis of rotation of the balls on the horizontal rod is much greater than the radii of the balls on the vertical rod.

Accepted Answer

(a) Calculate the moment of inertia of the baton when oriented as in Figure 8.24.

Apply Equation 8.12,

[latex]I ≡ \sum mr^2[/latex] [8.12]

neglecting the mass of the connecting rods:

[latex]\begin{aligned}I=& \sum m r^{2}=m_{1} r_{1}^{2}+m_{2} r_{2}^{2}+m_{3} r_{3}^{2}+m_{4} r_{4}^{2} \=&(0.20 \mathrm{~kg})(0.50 \mathrm{~m})^{2}+(0.30 \mathrm{~kg})(0.50 \mathrm{~m})^{2} \&+(0.20 \mathrm{~kg})(0.50 \mathrm{~m})^{2}+(0.30 \mathrm{~kg})(0.50 \mathrm{~m})^{2} \I=& 0.25 \mathrm{~kg} \cdot \mathrm{m}^{2}\end{aligned}[/latex]

(b) Calculate the moment of inertia of the baton when oriented as in Figure 8.25.

Apply Equation [latex]8.12[/latex] again, neglecting the radii of the [latex]0.20-\mathrm{kg}[/latex] balls.

[latex]\begin{aligned}I=& \sum m r^{2}=m_{1} r_{1}^{2}+m_{2} r_{2}^{2}+m_{3} r_{3}^{2}+m_{4} r_{4}^{2} \=&(0.20 \mathrm{~kg})(0)^{2}+(0.30 \mathrm{~kg})(0.50 \mathrm{~m})^{2}+(0.20 \mathrm{~kg})(0)^{2} \&+(0.30 \mathrm{~kg})(0.50 \mathrm{~m})^{2} \I=& 0.15 \mathrm{~kg} \cdot \mathrm{m}^{2}\end{aligned}[/latex]

REMARKS The moment of inertia is smaller in part (b) because in this configuration the [latex]0.20[/latex]-kg balls are essentially located on the axis of rotation.

Question 8.12: THE BATON TWIRLER GOAL Calculate a moment of inertia. PROBLE...

Related Answered Questions

Verified Answer:

Verified Answer:

Verified Answer:

BLOCKS AND PULLEY GOAL Solve a system requiring rotation concepts and the work-energy theorem. PROBLEM Two blocks with masses m1=5.00 kg and m2=7.00 kg are attached by a string as in Figure 8.31a, over a pulley with mass M=2.00 kg. The pulley, which turns on a frictionless axle, is a hollow ...

Verified Answer:

THE MERRY-GO-ROUND GOAL Apply conservation of angular momentum while combining two moments of inertia. PROBLEM A merry-go-round modeled as a disk of mass M=1.00×10² kg and radius R=2.00 m is rotating in a horizontal plane about a frictionless vertical axle (Fig. 8.37 is an overhead view of the ...

Verified Answer:

Verified Answer:

WALKING A HORIZONTAL BEAM GOAL Apply the two conditions of equilibrium. PROBLEM A uniform horizontal beam 5.00 m long and weighing 3.00×10² N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53.0∘ with the ...

Verified Answer:

DON’T CLIMB THE LADDER GOAL Apply the two conditions of equilibrium. PROBLEM A uniform ladder 10.0 m long and weighing 50.0 N rests against a frictionless vertical wall as in Figure 8.17a. If the ladder is just on the verge of slipping when it makes a 50.0∘ angle with the ground, find the ...

Verified Answer:

Verified Answer:

Verified Answer: