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Solutions to Assignment #3

LAW OF REACTION, Solutions (p. 129)

#4. At what angle should the banking be for an indoor running track to produce the fastest sprint times if the sprinters run the bend at 11.00 m/s and the track has semicircular curves of 50.0 metre circumference? First, compute the radius of curvature from the circumference of the semi-circle: m 915.1550.0/πθ/ === sr Next, from example 6.4.2 we can define the banking angle (θ) as:

775.37)775.0(tan

126.156121tan

)81.9(915.1511tantanθ

11

21

21

==⎟⎠⎞

⎜⎝⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−−

−−

rgvt

banking

The optimal banking angle for a sprinter running at 11.0 m/s is 37.8 deg.

#10. What is the centripetal force of a person swinging a 4.80 kg bucket if the radius of curvature of the path is 2.20 m and the angular speed is 4.65 rad/s? Centripetal force is defined as:

33.228

6225.21)20.2(80.465.4)2.2(80.4

/2

22

===

=ω== rmvmrmaF trlcentripeta

Thus, the centripetal force is 228 N.

LINEAR IMPULSE AND MOMENTUM, Solutions (p. 140)

#4. What horizontal impulse was produced off the blocks by a 50.0 kg swimmer whose horizontal velocity just before entering the water was 5.00 m/s?

2500)00.5(0.50

Impulse

=−=

−=∆= initialfinal mvmvmv

The horizontal impulse was 250 kg.m/s (or N.s)

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#6. What is the takeoff velocity of a 62.0 kg person who performs a standing broad jump and applies a gross impulse of 1000 N.s vertically and 250 N.s horizontally over a 1.100 s interval?

032.40.62/2500

/)Impulse(

=+=

= mvv xinitialxfinalx

3380.5958.3300.62/]042.6691000[

0.62/]100.1)81.9(0.621000[0

/)Impulse(

==−=−+=

−= mmgtvv yinitialyfinaly

The takeoff velocity is (4.03, 5.34) m/s. #12. If a 200 gram hockey puck is shot at 40.0 m/s and impacts a goalie’s pads for 100.0 ms, what average force will be experienced by the pads?

0801000008Impulse/t

00.8)0.40(200.00

Impulse

../.F

mvmvtF initialfinal

−=−==

−=−=

−==

The average force is −80.0 newtons.

ANGULAR IMPULSE AND MOMENTUM, Solutions (p. 145)

#2. What is the angular momentum of a 10.00 kg.m2 flywheel that is rotating at the rate of 23.0 rad/s? 230)10.00(23.0ω === IL The angular momentum is 230 kg.m2/s (N.m.s is OK). #6. What is the angular impulse produced by a hand crank that is producing a torque of 35.8 N.m for 5.00 minutes? 107400.60)00.5(8.35impulse ang. === tM The angular impulse is 10 740 N.m.s (kg.m2/s is OK) or 10.74 kN.m.s.

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CONSERVATION OF MOMENTUM, Solutions (p. 153) #2. From the previous question, if a constant force was applied at 20.0 cm from the axis of rotation for 0.400 s (assuming an initial angular velocity of zero and the same moment of inertia as at takeoff, 2.00 kg.m2) calculate: (a) the average force applied from rest to takeoff (b) the average angular acceleration from rest to takeoff

(c) the average angular accelerations for the ascending and descending phases of the jump

(a)

500)400.0(20.0

0.40

0.40)(

0.40)0.20(00.2

==×

=

==

==ω=

tdMF

tdFtM

IL takeofftakeofftakeoff

The average force applied is 500 newtons. (b)

0.5000.2

)200.0(500/ ====αIdFIM

Average angular acceleration is 50 rad/s2. (c)

714.9

½(0.700)0.306.26

½

57.28½(0.700)

0.200.30½

−=−

=ω−ω

=α

=−

=ω−ω

=α

flight

peaklanddescending

flight

takeoffpeakascending

t

t

The ascending angular acceleration is 28.6 rad/s2; descending is -9.71 rad/s2. #6. If a person leaves a diving board with an angular momentum of 6.65 kg.m2/s and a moment of inertia of 17.00 kg.m2, how much time will elapse before one rotation is completed?

062.163912.0/2/39118.000.17/65.6/

=π=ωθ====ω

tIL

The time taken to complete one revolution (2π radians) is 16.06 seconds.