14. Moments and Equilibrium - Versionshistorik

Ian den 20 september 2009 kl. 17.46

Ian — Sun, 20 Sep 2009 17:46:31 GMT

← Äldre version		Versionen från 20 september 2009 kl. 17.46
Rad 13:		Rad 13:

	• The resultant force on a body must be zero.		• The resultant force on a body must be zero.
-

	'''And'''		'''And'''
Rad 20:		Rad 19:


-	Example 14.1	+	'''[[Example 14.1]]'''

	A uniform beam has length 8 m and mass 60 kg. It is suspended by two ropes, as shown in the diagram below.		A uniform beam has length 8 m and mass 60 kg. It is suspended by two ropes, as shown in the diagram below.
Rad 28:		Rad 27:
	Find the tension in each rope.		Find the tension in each rope.

-	Solution	+
		+	'''Solution'''

	The diagram shows the forces acting on the beam.		The diagram shows the forces acting on the beam.
Rad 59:		Rad 59:


-	Example 14.2	+
		+	'''[[Example 14.2]]'''

	A beam, of mass 50 kg and length 5 m, rests on two supports as shown in the diagram. Find the magnitude of the reaction force exerted by each support.		A beam, of mass 50 kg and length 5 m, rests on two supports as shown in the diagram. Find the magnitude of the reaction force exerted by each support.
Rad 67:		Rad 68:
	Find the maximum mass that could be placed at either end of the beam if it is to remain in equilibrium.		Find the maximum mass that could be placed at either end of the beam if it is to remain in equilibrium.

-	Solution	+
		+	'''Solution'''

	The diagram shows the forces acting on the beam.		The diagram shows the forces acting on the beam.
Rad 126:		Rad 128:
	is 12.5 kg.		is 12.5 kg.

-	Example 14.3	+
		+	'''[[Example 14.3]]'''

	A ladder, of length 3 m and mass 20 kg, leans against a smooth, vertical wall so that the angle between the horizontal ground and the ladder is 60.		A ladder, of length 3 m and mass 20 kg, leans against a smooth, vertical wall so that the angle between the horizontal ground and the ladder is 60.
Rad 132:		Rad 135:
	(b) Find the minimum value of the coefficient of friction between the ladder and the ground.		(b) Find the minimum value of the coefficient of friction between the ladder and the ground.

-	Solution	+
		+	'''Solution'''

	The diagram shows the forces acting on the ladder		The diagram shows the forces acting on the ladder

Ian den 20 september 2009 kl. 17.42

Ian — Sun, 20 Sep 2009 17:42:34 GMT

← Äldre version		Versionen från 20 september 2009 kl. 17.42
Rad 63:		Rad 63:
	A beam, of mass 50 kg and length 5 m, rests on two supports as shown in the diagram. Find the magnitude of the reaction force exerted by each support.		A beam, of mass 50 kg and length 5 m, rests on two supports as shown in the diagram. Find the magnitude of the reaction force exerted by each support.

-		+	[[Image:E14.2fig1.GIF]]
-		+
-		+

	Find the maximum mass that could be placed at either end of the beam if it is to remain in equilibrium.		Find the maximum mass that could be placed at either end of the beam if it is to remain in equilibrium.
Rad 73:		Rad 71:
	The diagram shows the forces acting on the beam.		The diagram shows the forces acting on the beam.

-		+	[[Image:E14.2fig2.GIF]]
-		+
-		+
-		+
-		+
-		+

	Taking moments about the point where R1 acts gives:		Taking moments about the point where R1 acts gives:
Rad 109:		Rad 102:
	.		.

-		+	[[Image:E14.2fig3.GIF]]
-		+
-		+
-		+
-		+
-		+

	Taking moments about the point where R1 acts gives:		Taking moments about the point where R1 acts gives:
Rad 127:		Rad 115:
	Similarly for a mass placed at the right hand end of the beam:		Similarly for a mass placed at the right hand end of the beam:

-		+	[[Image:E14.2fig4.GIF]]
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+

	<math>\begin{align}		<math>\begin{align}
Rad 154:		Rad 134:
	Solution		Solution

-	The diagram shows the forces acting on the ladder.	+	The diagram shows the forces acting on the ladder
-		+
-		+
-		+
-		+
-		+
-		+
-		+

		+	[[Image:E14.3fig1.GIF]]

	(a)		(a)

Ian den 20 september 2009 kl. 17.17

Ian — Sun, 20 Sep 2009 17:17:41 GMT

← Äldre version		Versionen från 20 september 2009 kl. 17.17
Rad 24:		Rad 24:
	A uniform beam has length 8 m and mass 60 kg. It is suspended by two ropes, as shown in the diagram below.		A uniform beam has length 8 m and mass 60 kg. It is suspended by two ropes, as shown in the diagram below.

-		+	[[Image:E14.1fig1.GIF]]
-		+
-		+

	Find the tension in each rope.		Find the tension in each rope.
Rad 34:		Rad 32:
	The diagram shows the forces acting on the beam.		The diagram shows the forces acting on the beam.

-		+	[[Image:E14.1fig2.GIF]]
-		+
-		+
-		+
-		+
-		+

	Take moments about the point where T1 acts to give:		Take moments about the point where T1 acts to give:
-

	<math>\begin{align}		<math>\begin{align}

Ian den 20 september 2009 kl. 16.50

Ian — Sun, 20 Sep 2009 16:50:01 GMT

← Äldre version		Versionen från 20 september 2009 kl. 16.50
Rad 6:		Rad 6:
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|
	\|}		\|}
		+
		+
		+	Key Points
		+
		+	'''In equilibrium''':
		+
		+	• The resultant force on a body must be zero.
		+
		+
		+	'''And'''
		+
		+	• The resultant moment on a body must be zero.
		+
		+
		+	Example 14.1
		+
		+	A uniform beam has length 8 m and mass 60 kg. It is suspended by two ropes, as shown in the diagram below.
		+
		+
		+
		+
		+
		+	Find the tension in each rope.
		+
		+	Solution
		+
		+	The diagram shows the forces acting on the beam.
		+
		+
		+
		+
		+
		+
		+
		+
		+	Take moments about the point where T1 acts to give:
		+
		+
		+	<math>\begin{align}
		+	& 5\times {{T}_{2}}=3\times 588 \\
		+	& {{T}_{2}}=\frac{3\times 588}{5}=352.8=353\text{ N (to 3sf)} \\
		+	\end{align}</math>
		+
		+
		+	Take moments about the point where T2 acts to give:
		+
		+
		+	<math>\begin{align}
		+	& 5\times {{T}_{1}}=2\times 588 \\
		+	& {{T}_{1}}=\frac{2\times 588}{5}=235.2=235\text{ N (to 3sf)} \\
		+	\end{align}</math>
		+
		+
		+	Finally for vertical equilibrium we require
		+	<math>{{T}_{1}}+{{T}_{2}}=588</math>
		+	,which can be used to check the tensions. In is the case we have:
		+
		+	<math>352.8+235.2=588</math>
		+
		+
		+
		+	Example 14.2
		+
		+	A beam, of mass 50 kg and length 5 m, rests on two supports as shown in the diagram. Find the magnitude of the reaction force exerted by each support.
		+
		+
		+
		+
		+
		+	Find the maximum mass that could be placed at either end of the beam if it is to remain in equilibrium.
		+
		+	Solution
		+
		+	The diagram shows the forces acting on the beam.
		+
		+
		+
		+
		+
		+
		+
		+
		+	Taking moments about the point where R1 acts gives:
		+
		+
		+	<math>\begin{align}
		+	& 2\times {{R}_{2}}=1.5\times 490 \\
		+	& {{R}_{2}}=\frac{1.5\times 490}{2}=367.5=368\text{ N (to 3sf)} \\
		+	\end{align}</math>
		+
		+
		+	Taking moments about the point where R2 acts gives:
		+
		+
		+	<math>\begin{align}
		+	& 2\times {{R}_{1}}=0.5\times 490 \\
		+	& {{R}_{1}}=\frac{0.5\times 490}{2}=122.5=123\text{ N (to 3sf)} \\
		+	\end{align}</math>
		+
		+
		+	For vertical equilibrium we require
		+	<math>{{R}_{1}}+{{R}_{2}}=490</math>
		+	, which can be used to check the tensions. In is the case we have:
		+
		+	<math>367.5+122.5=490</math>
		+
		+
		+	First consider the greatest mass that can be placed at the left hand end of the beam. The diagram below shows the extra force that must now be considered. When the maximum possible mass is used,
		+	<math>{{R}_{2}}=0</math>
		+	.
		+
		+
		+
		+
		+
		+
		+
		+
		+	Taking moments about the point where R1 acts gives:
		+
		+
		+	<math>\begin{align}
		+	& 1\times mg=1.5\times 490 \\
		+	& m=\frac{1.5\times 490}{9.8}=75\text{ kg} \\
		+	\end{align}</math>
		+
		+
		+	Similarly for a mass placed at the right hand end of the beam:
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+	<math>\begin{align}
		+	& 2\times mg=0.5\times 490 \\
		+	& m=\frac{0.5\times 490}{2g}=12.5\text{ kg} \\
		+	\end{align}</math>
		+
		+
		+	Hence the greatest mass that can be placed at either end of the beam
		+	is 12.5 kg.
		+
		+	Example 14.3
		+
		+	A ladder, of length 3 m and mass 20 kg, leans against a smooth, vertical wall so that the angle between the horizontal ground and the ladder is 60.
		+	(a) Find the magnitude of the friction and normal reaction forces that act on the ladder, if it is in equilibrium.
		+	(b) Find the minimum value of the coefficient of friction between the ladder and the ground.
		+
		+	Solution
		+
		+	The diagram shows the forces acting on the ladder.
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+	(a)
		+	Considering the horizontal forces gives:
		+
		+
		+	<math>F=S</math>
		+
		+
		+	Considering the vertical forces gives:
		+
		+
		+	<math>R=196</math>
		+
		+
		+	Taking moment about the base of the ladder gives:
		+
		+
		+	<math>\begin{align}
		+	& 196\times 1.5\cos 60{}^\circ =S\times 3\sin 60{}^\circ \\
		+	& S=\frac{196\times 1.5\cos 60{}^\circ }{3\sin 60{}^\circ }=\frac{196\times \cos 60{}^\circ }{2\sin 60{}^\circ }=\frac{196}{2\tan 60{}^\circ }=56.6\text{ N (to 3 sf)} \\
		+	\end{align}</math>
		+
		+
		+	But since
		+
		+	<math>F=S</math>
		+	, the friction force has magnitude 56.6 N.
		+
		+	(b)
		+
		+	Using the friction inequality,
		+
		+	<math>F\le \mu R</math>
		+	gives:
		+
		+
		+
		+	<math>\begin{align}
		+	& \frac{196}{2\tan 60{}^\circ }\le \mu \times 196 \\
		+	& \mu \ge \frac{1}{2\tan 60{}^\circ } \\
		+	& \mu \ge 0.289\text{ (to 3sf)} \\
		+	\end{align}</math>

Ian den 20 september 2009 kl. 16.02

Ian — Sun, 20 Sep 2009 16:02:58 GMT

← Äldre version		Versionen från 20 september 2009 kl. 16.02
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-	{{Selected tab\|[[13. Moments\|Theory]]}}	+	{{Selected tab\|[[14. Moments and Equilibrium\|Theory]]}}
-	{{Not selected tab\|[[13. Exercises\|Exercises]]}}	+	{{Not selected tab\|[[14. Exercises\|Exercises]]}}
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Ian: New page: NOTOC {| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | style="border-bottom:1px solid #797979" width="5px" | {{Selected tab|Theory}} {{No...

Ian — Sun, 20 Sep 2009 15:58:52 GMT

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