PHL246H1 Midterm: Solutions to ch1 for test 1 (30%).

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orchidfox628

11 Apr 2011

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Chapter one: for the truth table, go to http://turner. faculty. swau. edu/mathematics/materialslibrary/truth/ and type in ~(a&(b+~c)) for the boolean expression and hit generate table. A truth table will have 2 to the power of x rows, where x is the number of letters/variables you are using. (a+b)&(a+c) for the right: if b1,b2, ,bn are mutually exclusive, then only one of them can be true at one point in time. Now, in a conjunction (eg a^b1), if any term is false, then the whole conjunction is false. So, if only one of b1 through bn can be true at any point in time, then all but one are false at the point in time. It follows then that all but one of the conjunctions a^bx (where bx will just be the whole set of b1 through bn) must be false when any single sentence in bx is true. For example, if b1 is true, b2 through bn is false, and therefore.

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1) A stepladder (shown in figure 12-52 on page 325 in your book)has sides which are each 3.0m long and hinged at the top. A 0.8mtie-rod is located half way up. A man weighing 800N climbs 2.1malong the ladderâs length. So that weâre all using the samevariables, let us define
W= 800m L=0.3 m d=1.2m s=8.0
Assuming that the floor is frictionless and neglecting the mass ofthe ladder â¦
(0.5 points)
Write an expression for the angle that the ladder makes with thehorizontal.

Model the situation with Newtonâs Second Law [and its rotationalequivalent]. Set up the system of equations as if you were to solvefor the quantities below. You equations should be in terms of thevariables given above only (no numbers) and trig functions of theangle above (just call it ).

One last comment; in addition to the forces below, thereâs one more(your job to figure out what it is) that will be in your equations.The force has components. Tip: Rather than defining a new angle andmessing with more trig functions, just call the x-component Fx andthe y-component Fy in your equations.

solve the above equations for the following quantities (give me anexpression in terms of the variables first, then give me the actualnumber)â¦
(1 points)
a) The tension in the tie-rod. (Call it T in your free-bodydiagrams and equations.)
The magnitude of the force on the ladder from the floor on;
(1 points)
b) the left side of the ladder (where the person is)? (Call it FLin your FBDs and your equations.)
(1 points)
c) the right side of the ladder? (Call it FR in your FBDs and yourequations.)
***Remember to draw free body diagrams (for yourself, not to besubmitted) and write out all equations using variables before youplug in numbers. Also notice that the numbers in your book areslightly different than those in this problem, but the bookâsanswer should provide you a way to check your method andsolution.

Rotational motion with a constant nonzero acceleration is not uncommon in the world around us. For instance, many machines have spinning parts. When the machine is turned on or off, the spinning parts tend to change the rate of their rotation with virtually constant angular acceleration. Many introductory problems in rotational kinematics involve the motion of a particle with constant, nonzero angular acceleration. The kinematic equations for such motion can be written as

$\theta (t) = \theta_0 +\omega_0t + \frac{1}{2}\alpha t^2$

and

$\omega (t) = \omega_0 + \alpha t$ .

Here, the symbols are defined as follows:

is the angular position of the particle at time .
is the initial angular position of the particle.
is the angular velocity of the particle at time .
is the initial angular velocity of the particle.
is the angular acceleration of the particle.
is the time that has elapsed since the particle was located at its initial position.

In answering the following questions, assume that the angular acceleration is constant and nonzero: $\alpha \neq 0$ .

True or false: The quantity represented by

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by theta_0

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by omega_0

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by omega

is a function of time (i.e., is not constant).

	true
	false

Which of the following equations is not an explicit function of time

? Keep in mind that an equation that is an explicit function of time involves

as a variable.

	$\theta=\theta_0+\omega_0 t+\frac{1}{2}\alpha t^2$
	$\omega=\omega_0+\alpha t$
	$\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)$

In the equation $\omega=\omega_0+\alpha t$ , what does the time variable

represent?

Choose the answer that is always true. Several of the statements may be true in a particular problem, but only one is always true.

	the moment in time at which the angular velocity equals $\omega_0$
	the moment in time at which the angular velocity equals $\omega$
	the time elapsed from when the angular velocity equals $\omega_0$ until the angular velocity equals $\omega$

ceruleanbutterfly285

to change the rate of their rotation with virtually constant angular acceleration. Many introductory problems in rotational kinematics involve the motion of a particle with constant, nonzero angular acceleration. The kinematic equations for such motion can be written as

$\theta (t) = \theta_0 +\omega_0t + \frac{1}{2}\alpha t^2$

and

$\omega (t) = \omega_0 + \alpha t$ .

Here, the symbols are defined as follows:

is the angular position of the particle at a time t.
is the initial angular position of the particle.
is the angular velocity of the particle at a time t.
is the initial angular velocity of the particle.
is the angular acceleration of the particle.
is the time that has elapsed since the particle was located at its initial position.

In answering the following questions, assume that the angular acceleration is constant and nonzero: $\alpha \neq 0$ .

True or false: The quantity represented by theta

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by theta_0

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by omega_0

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by omega

is a function of time (i.e., is not constant).

	true
	false

Which of the following equations is not an explicit function of time

? Keep in mind that an equation that is an explicit function of time involves

as a variable.

	$\theta=\theta_0+\omega_0 t+\frac{1}{2}\alpha t^2$
	$\omega=\omega_0+\alpha t$
	$\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)$

In the equation $\omega=\omega_0+\alpha t$ , what does the time variable

represent?

Choose the answer that is always true. Several of the statements may be true in a particular problem, but only one is always true.

	the moment in time at which the angular velocity equals $\omega_0$
	the moment in time at which the angular velocity equals $\omega$
	the time elapsed from when the angular velocity equals $\omega_0$ until the angular velocity equals $\omega$

Consider two particles A and B. The angular position of particle A, with constant angular acceleration, depends on time according to $\theta_{\rm A}(t)=\theta_0+\omega_0t+\frac{_1}{^2}\alpha t^2$ . At time $t=t_1$ , particle B, which also undergoes constant angular acceleration, has twice the angular acceleration, half the angular velocity, and the same angular position that particle A had at time $t=0$ .

Which of the following equations describes the angular position of particle B?

	$\theta_{\rm B}(t)=\theta_0+2\omega_0t+\frac{1}{4}\alpha t^2$
	$\theta_{\rm B}(t)=\theta_0+\frac{1}{2}\omega_0t+\alpha t^2$
	$\theta_{\rm B}(t)=\theta_0+2\omega_0(t-t_1) +\frac{1}{4}\alpha (t-t_1)^2$
	$\theta_{\rm B}(t)=\theta_0+\frac{1}{2}\omega_0(t-t_1)+\alpha (t-t_1)^2$
	$\theta_{\rm B}(t)=\theta_0+2\omega_0(t+t_1) +\frac{1}{4}\alpha (t+t_1)^2$
	$\theta_{\rm B}(t)=\theta_0+\frac{1}{2}\omega_0(t+t_1)+\alpha (t+t_1)^2$

How long after the time t_1

does the angular velocity of particle B equal that of particle A?

	$\frac{\omega_0}{4\alpha}$
	$\frac{\omega_0+4\alpha t_1}{2\alpha}$
	$\frac{\omega_0+2\alpha t_1}{2\alpha}$
	The two particles never have the same angular velocity.

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PHL246H1 Midterm: Solutions to ch1 for test 1 (30%).

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