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Take-home experiment

Plan an experiment to analyze changes to a system's angular momentum. Choose a system capable of rotational motion such as a lazy Susan or a merry-go-round. Predict how the angular momentum of this system will change when you add an object to the lazy Susan or jump onto the merry-go-round. What variables can you control? What are you measuring? In other words, what are your independent and dependent variables? Are there any independent variables that it would be useful to keep constant (angular velocity, perhaps)? Collect data in order to calculate or estimate the angular momentum of your system when in motion. What do you observe? Collect data in order to calculate the change in angular momentum as a result of the interaction you performed.

Using your data, how does the angular momentum vary with the size and location of an object added to the rotating system?

Calculating the torque in a kick

The person whose leg is shown in [link] kicks his leg by exerting a 2000-N force with his upper leg muscle. The effective perpendicular lever arm is 2.20 cm. Given the moment of inertia of the lower leg is 1.25 kg m 2 size 12{1 "." "25"`"kg" cdot m rSup { size 8{2} } } {} , (a) find the angular acceleration of the leg. (b) Neglecting the gravitational force, what is the rotational kinetic energy of the leg after it has rotated through 57 . size 12{"57" "." 3`°} {} (1.00 rad)?

The figure shows a human leg, from the thighs to the feet which is bent at the knee joint. The radius of curvature of the knee is indicated as r equal to two point two zero centimeters and the moment of inertia of the lower half of the leg is indicated as I equal to one point two five kilogram meter square. The direction of torque is indicated by a red arrow in anti-clockwise direction, near the knee.
The muscle in the upper leg gives the lower leg an angular acceleration and imparts rotational kinetic energy to it by exerting a torque about the knee. F is a vector that is perpendicular to r . This example examines the situation.

Strategy

The angular acceleration can be found using the rotational analog to Newton's second law, or α = net τ / I size 12{α="net "τ/I} {} . The moment of inertia I size 12{I} {} is given and the torque can be found easily from the given force and perpendicular lever arm. Once the angular acceleration α size 12{α} {} is known, the final angular velocity and rotational kinetic energy can be calculated.

Solution to (a)

From the rotational analog to Newton's second law, the angular acceleration α size 12{α} {} is

α = net τ I . size 12{α= { {"net "τ} over {I} } } {}

Because the force and the perpendicular lever arm are given and the leg is vertical so that its weight does not create a torque, the net torque is thus

net τ = r F = 0 . 0220 m 2000 N = 44 . 0 N m.

Substituting this value for the torque and the given value for the moment of inertia into the expression for α size 12{α} {} gives

α = 44 . 0 N m 1 . 25 kg m 2 = 35 . 2 rad/s 2 . size 12{α= { {"44" "." 0" N" cdot m} over {1 "." "25"" kg" cdot m rSup { size 8{2} } } } ="35" "." 2" rad/s" rSup { size 8{2} } } {}

Solution to (b)

The final angular velocity can be calculated from the kinematic expression

ω 2 = ω 0 2 + 2 αθ

or

ω 2 = 2 αθ size 12{ω rSup { size 8{2} } =2 ital "αθ"} {}

because the initial angular velocity is zero. The kinetic energy of rotation is

KE rot = 1 2 2 size 12{"KE" rSub { size 8{"rot"} } = { {1} over {2} } Iω rSup { size 8{2} } } {}

so it is most convenient to use the value of ω 2 size 12{ω rSup { size 8{2} } } {} just found and the given value for the moment of inertia. The kinetic energy is then

KE rot = 0.5 1 .25 kg m 2 70. 4 rad 2 / s 2 = 44 . 0 J .

Discussion

These values are reasonable for a person kicking his leg starting from the position shown. The weight of the leg can be neglected in part (a) because it exerts no torque when the center of gravity of the lower leg is directly beneath the pivot in the knee. In part (b), the force exerted by the upper leg is so large that its torque is much greater than that created by the weight of the lower leg as it rotates. The rotational kinetic energy given to the lower leg is enough that it could give a ball a significant velocity by transferring some of this energy in a kick.

Practice Key Terms 2

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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