A uniform sphere of mass 2.50 kg and radius 15.0 cm is released from rest at the top of an incline that is 5.25 m long and makes an angle of $35^{\circ}$ with the horizontal. Assuming it rolls without slipping, (a) determine its total kinetic energy at the bottom of the incline. (b) Determine its rotational kinetic energy at the bottom of the incline. (c) What type of friction, static or kinetic, is acting on the surface of the sphere? Explain. (d) Determine the force of friction in part (d).

A car on a circular track accelerates from rest. (a) The car experiences (1) only angular acceleration, (2) only centripetal acceleration, (3) both angular and centripetal accelerations. Why? (b) If the radius of the track is 0.30 km and the magnitude of the constant angular acceleration is $4.5 \times 10^{-3}\ \mathrm{rad} / \mathrm{s}^{2},$ how long does the car take to make one lap around the track? (c) What is the total (vector) acceleration of the car when it has completed half of a lap?

Perform the operations, given $a = 3$, $b = −4$, and

$A=$

$$\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix}$$

, $B=$

$$\begin{bmatrix} 0 & 1 \\ -1 & 2 \\ \end{bmatrix}$$

, $O=$

$$\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix}$$

Evaluate

$$A+B$$

Find the intervals of concavity and the inflection points. $f(x)=\frac{1}{1-x^{2}}$

Determine whether the statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.

A $4 \times 7$ matrix has four columns.

Do all points on a wheel rotating about a fixed axis through its center have the same angular velocity? The same tangential speed? Explain.

Find, if possible, A + B.

$$A=\begin{bmatrix} {1} & {2} \\ {2} & {1} \end{bmatrix},\ B=\begin{bmatrix} {-3} & {-2} \\ {4} & {2} \end{bmatrix}$$

The emissivity of an object is 0.50. (a) Compared with a perfect blackbody at the same temperature, this object would radiate (1) more power, (2) the same amount of power, (3) less power. Why? (b) Calculate the ratio of the power radiated by the blackbody to that radiated by the object.

Find the solution of the differential equation that satisfies the given initial condition. $\(\frac{d P}{d t}=\sqrt{P t}, \quad P(1)=2\)$

Using the given matrices, calculate the quantities, if they are defined. 7K - H

$$A=\left[\begin{array}{ll} {7} & {0} \\ {1} & {9} \end{array}\right] \quad B=\left[\begin{array}{ll} {0} & {2} \\ {13} & {6} \end{array}\right] \quad C=\left[\begin{array}{ll} {15} & {0} \\ {0} & {2} \end{array}\right]$$

$$E=\left[\begin{array}{lll} {x} & {5} & {3} \\ {0} & {9} & {1} \\ {6} & {4} & {0} \end{array}\right] \quad F=\left[\begin{array}{lll} {y} & {0} & {y} \\ {0} & {1} & {0} \\ {0} & {1} & {1} \end{array}\right]$$

$$G=\left[\begin{array}{lll} {1} & {2} & {3} \\ {5} & {4} & {6} \\ {9} & {7} & {8} \end{array}\right]$$

$$H=\left[\begin{array}{lll} {2} & {1} & {3} \\ {3} & {7} & {6} \end{array}\right] \quad K=\left[\begin{array}{ll} {9} & {2} \\ {1} & {5} \\ {7} & {8} \end{array}\right]$$

Express the following signals in terms of singularity functions.

$$\text { (a) } v(t)=\left\{\begin{array}{cc}{0,} & {t<0} \\ {-5,} & {t>0}\end{array}\right.$$

$$\text { (b) } i(t)=\left\{\begin{array}{cl}{0,} & {t<1} \\ {-10,} & {1<t<3} \\ {10,} & {3<t<5} \\ {0,} & {t>5}\end{array}\right.$$

$$\text { (c) } x(t)=\left\{\begin{array}{ll}{t-1,} & {1<t<2} \\ {1,} & {2<t<3} \\ {4-t,} & {3<t<4} \\ {0,} & {\text { Otherwise }}\end{array}\right.$$

$$\text { (d) } y(t)=\left\{\begin{array}{cc}{2,} & {t<0} \\ {-5,} & {0<t<1} \\ {0,} & {t>1}\end{array}\right.$$

Compare the functions $f(x)=3 x+2,$ $g(x)=2 x^{2}+3,$ and $h(x)=2^{x}$. Show that as x increases, h(x) will eventually exceed f(x) and g(x).

Both alkyl- and arylamins have a low barrier for pyramidal inversion at nitrogen, which prevents the separation of chiral amines into their enantiomers. The barrier for inversion at nitrogen in alkylamines is approximately 25 kJ/mol (6 kcal/mol), whereas for arylamines it is much lower, on the order of 6.3 kJ/mol (1.5 kcal/mol). Can you suggest a reason for the difference?

The asymmetry in the binding energy curve accounts for thermal expansion of solids. How can information from the molecular spectra be used to determine the shape of this curve?