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Lexcie De Jesus

Lexcie D.

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A very small cube of mass $m$ is placed on the inside of a funnel rotating about a vertical axis at constant rate $a$ radian $/ \mathrm{sec}$. The wall of the funnel makes an angle $\theta$ with the horizontal, If the centre of the cube is at a distance $r$ from the axis of rotation of the funnel, what are the largest and the smallest values of $\omega$ for which the block neither slip down nor skid up? Solution If $v$ as well as $\omega$ is large, the block moves up and if $v$ is small, the block moves down. If $N$ is the normal reaction, $$ N-m g \cos \theta+\frac{m v^{2}}{r} \cos (90-\theta)-m g \cos \theta+\frac{m v^{2}}{r} \sin \theta $$ $\therefore$ The frictional force $=\mu \mathrm{N}=\mu m\left(g \cos \theta+v^{2} \sin \theta\right)$ This acts down because the cube has a tendency to move up when $\omega$ is large. The force must be just sufficient to balance the force up the funnel surface. 'lhis is $\frac{m v^{2}}{r} \cos \theta-m g \sin \theta$ $\therefore \mu m\left(g \cos \theta \mid \frac{v^{2}}{r} \sin \theta\right)=m\left(\frac{v^{2}}{r} \cos \theta \quad g \sin \theta\right)$ $\mu g \cos \theta+g \sin \theta-\frac{v^{2}}{r}(\cos \theta-\mu \sin \theta)$ $\therefore \quad v^{2}-\frac{g r(\mu \cos \theta \mid \sin \theta)}{\cos \theta-\mu \sin \theta} \quad \Rightarrow \quad v-\sqrt{\frac{g r(\mu \cos \theta \mid \sin \theta}{\cos \theta-\mu \sin \theta}}$ $\therefore \quad \omega^{\max }-\frac{v}{r}-\sqrt{\frac{g(\mu \cos \theta \mid \sin \theta)}{r(\cos \theta-\mu \sin \theta)}}$ When $\omega$ is small, $\mu\left(m g \cos \theta \times \frac{m v^{2}}{r} \sin \theta\right)$ acts upwards. $\therefore \mu\left(m g \cos \theta+\frac{m v^{2}}{r} \sin \theta\right)-m g \sin \theta+\frac{m v^{2}}{r} \cos \theta-0$or $g(\sin \theta \quad \mu \cos \theta)-\frac{v^{2}}{r}(\mu \sin \theta \mid \cos \theta)$ $\therefore \quad v^{2}-\frac{r g(\sin \theta-\mu \cos \theta)}{\mu \sin \theta+\cos \theta} \Rightarrow v-\sqrt{\frac{r g(\sin \theta-\mu \cos \theta)}{\mu \sin \theta+\cos \theta}}$ $\therefore \quad \omega^{\min }-\frac{v}{r}-\sqrt{\frac{g(\sin \theta-\mu \cos \theta)}{r(\mu \sin \theta+\cos \theta)}}$

A very small cube of mass $m$ is placed on the inside of a funnel rotating about a vertical axis at constant rate $a$ radian $/ \mathrm{sec}$. The wall of the funnel makes an angle $\theta$ with the horizontal, If the centre of the cube is at a distance $r$ from the axis of rotation of the funnel, what are the largest and the smallest values of $\omega$ for which the block neither slip down nor skid up? Solution If $v$ as well as $\omega$ is large, the block moves up and if $v$ is small, the block moves down. If $N$ is the normal reaction, $$ N-m g \cos \theta+\frac{m v^{2}}{r} \cos (90-\theta)-m g \cos \theta+\frac{m v^{2}}{r} \sin \theta $$ $\therefore$ The frictional force $=\mu \mathrm{N}=\mu m\left(g \cos \theta+v^{2} \sin \theta\right)$ This acts down because the cube has a tendency to move up when $\omega$ is large. The force must be just sufficient to balance the force up the funnel surface. 'lhis is $\frac{m v^{2}}{r} \cos \theta-m g \sin \theta$ $\therefore \mu m\left(g \cos \theta \mid \frac{v^{2}}{r} \sin \theta\right)=m\left(\frac{v^{2}}{r} \cos \theta \quad g \sin \theta\right)$ $\mu g \cos \theta+g \sin \theta-\frac{v^{2}}{r}(\cos \theta-\mu \sin \theta)$ $\therefore \quad v^{2}-\frac{g r(\mu \cos \theta \mid \sin \theta)}{\cos \theta-\mu \sin \theta} \quad \Rightarrow \quad v-\sqrt{\frac{g r(\mu \cos \theta \mid \sin \theta}{\cos \theta-\mu \sin \theta}}$ $\therefore \quad \omega^{\max }-\frac{v}{r}-\sqrt{\frac{g(\mu \cos \theta \mid \sin \theta)}{r(\cos \theta-\mu \sin \theta)}}$ When $\omega$ is small, $\mu\left(m g \cos \theta \times \frac{m v^{2}}{r} \sin \theta\right)$ acts upwards. $\therefore \mu\left(m g \cos \theta+\frac{m v^{2}}{r} \sin \theta\right)-m g \sin \theta+\frac{m v^{2}}{r} \cos \theta-0$or $g(\sin \theta \quad \mu \cos \theta)-\frac{v^{2}}{r}(\mu \sin \theta \mid \cos \theta)$ $\therefore \quad v^{2}-\frac{r g(\sin \theta-\mu \cos \theta)}{\mu \sin \theta+\cos \theta} \Rightarrow v-\sqrt{\frac{r g(\sin \theta-\mu \cos \theta)}{\mu \sin \theta+\cos \theta}}$ $\therefore \quad \omega^{\min }-\frac{v}{r}-\sqrt{\frac{g(\sin \theta-\mu \cos \theta)}{r(\mu \sin \theta+\cos \theta)}}$

Practice Problem in Physics for the JEE Main and Advanced

Work, Energy, Power and Circular Motion

Section A
Calculate the pH and the pOH of each of the following solutions at $25^{\circ} \mathrm{C}$ for which the substances ionize completely: (a) $0.000259 \mathrm{M} \mathrm{HClO}_{4}$ (b) 0.21 $M$ NaOH (c) $0.000071 M$ Ba(OH) $_{2}$ (d) $2.5 M \mathrm{KOH}$

Calculate the pH and the pOH of each of the following solutions at $25^{\circ} \mathrm{C}$ for which the substances ionize completely: (a) $0.000259 \mathrm{M} \mathrm{HClO}_{4}$ (b) 0.21 $M$ NaOH (c) $0.000071 M$ Ba(OH) $_{2}$ (d) $2.5 M \mathrm{KOH}$

Chemistry

If ocean water consisted of $3.5 \%$ salt, what would be the freezing point of an ocean?

If ocean water consisted of $3.5 \%$ salt, what would be the freezing point of an ocean?

General Chemistry: Principles and Modern Applications

Questions asked

INSTANT ANSWER

A box with mass M = 14.3 kg is placed on a rough surface with coefficient of kinetic friction of 0.245. The mass is attached to two balls m1 and m2 = 10.6 [kg] via two pieces of rope and the system starts sliding as shown. The rope slides around the pulley mechanism frictionlessly. Given that the first rope exerts a tension force of 112 [N] on both the box and m1, what is the mass of m1? a. Draw the FBD of each object. b. Apply Newton’s second law on each object. c. Give the expression for m1 before substituting the numerical values.

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INSTANT ANSWER

FGURE 9 Geolocic cross section for relatve age analysis. Place etters on the lines along the right sicle of the cross section to indicate the relative aces of the rock units and other features (uncorformities, fault). from oldest (first) to yourcest (dast). 17. F [roungest| 16. R 15. 14. 13. 12. 11. 10. 9. 8. 7. 6. 5. 4. 3. 2. 1. \{oldest| 15. \{roungest| 14. 13. 12. 11. 10. 9. 3. 7. 6. 5. 4. 3. 2. 1. \{oldest| ACOURE 10 Geobgic cross section of the Grand Canyon for relative age andysis Place leflers on the lires abong the right side of the cross section to indic ate the relative ages of the rock unts and urcorformities, from oldest (first) to yourcest (ast).

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ANSWERED

Daniel Gosser verified

Numerade educator

1) What is the pOH of the solution formed by mixing 15 mL of 0.10 M HNO₃ and 10 mL 0.15 M Ca(OH)₂? Express answer in 2 decimal places. 2) Which of the following indicates the most basic solution at 25oC? Select one: a. [OH⁻] = 7 × 10⁻⁵ M b. pOH = 6.7 c. pH = 4.2 d. [H⁺] = 1 × 10⁻¹⁰ M

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ANSWERED

Shalini Tyagi verified

Numerade educator

If ocean water consisted of 3.5% salt (NaCl), what would be the freezing point of an ocean? (Kf H2O = 1.86°C m-1) Select one: A. 0.00°C B. -1.88°C C. -0.651°C D. -2.23°C

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ANSWERED

David Collins verified

Numerade educator

In a beaker, 25.0 mL each of organic compounds, X and Y, were mixed. The total volume was measured to be 48.5 mL. Which of the following can be drawn about compounds X and Y? Choose all that are possible. Select one or more: A. Either of compounds X or Y is polar while the other is non-polar. B. Compound Y molecules have stronger attractive forces to one another than compound X molecules. C. Compounds X and Y are more miscible to one another at lower temperature. D. The beaker feels warm as mixing occurs. E. For compound X molecules to interact with compound Y molecules, energy is absorbed.

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ANSWERED

David Collins verified

Numerade educator

Consider the diagram on the left: 1.00 g H2(g) is maintained at 1 atm pressure in a cylinder closed off by a freely moving piston. Which sketch on the right best represents the mixture obtained when 1.00 g He(g) is added?

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ANSWERED

David Collins verified

Numerade educator

A handbook lists the normal boiling point of isooctane, a gasoline component, as 99.2ºC and its enthalpy of vaporization as 35.76 kJ mol-1. Calculate the vapor pressure of isooctane at 25ºC. Select one: A.0.0564 atm B.0.00133 atm C.128 atm D.17.7 atm

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ANSWERED

Shalini Tyagi verified

Numerade educator

One of the go-to health products today is rubbing alcohol. A commercial brand reports that a 1.00 L gallon of its rubbing alcohol contains 70% (v/v) isopropyl alcohol (MM = 60.1, d = 0.786 g/mL). Based from these given, determine the following variables. Assume that solvent is distilled H2O (MM = 18.016, d = 1.00 g/mL). Express all answers in three significant figures. A. Mole fraction of isopropyl alcohol in the gallon: Answer B. Molarity of isopropyl alcohol in the gallon: Answer M C. Volume of isopropyl alcohol in a 50.0 mL aliquot drawn from the gallon: Answer mL

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ANSWERED

David Collins verified

Numerade educator

33.5 1.00 P atm 0.127 63.1 63.2 77.4 126.0 T Kelvin

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INSTANT ANSWER

1. One of the go-to health products today is rubbing alcohol. A commercial brand reports that a 1.00 L gallon of its rubbing alcohol contains 70% (v/v) isopropyl alcohol (MM = 60.1, d = 0.786 g/mL). Based from these given, determine the following variables. Assume that solvent is distilled H2O (MM = 18.016, d = 1.00 g/mL). Express all answers in three significant figures. A. Mole fraction of isopropyl alcohol in the gallon: Answer B. Molarity of isopropyl alcohol in the gallon: Answer M C. Volume of isopropyl alcohol in a 50.0 mL aliquot drawn from the gallon: Answer mL 2) How many liters of H2(g) at STP are produced per gram of Al(s) consumed in the following reaction? Al(s) + HCl(aq) → AlCl3(aq) + H2(g) A. 0.830 L B. 1.25 L C. 1.36 L D. 0.907 L 3) A 1.75 m aqueous solution boils at 101.1°C. Which of the following would be the best identity of solute? (Kb H2O = 0.514°C m-1) A.CH3COOH B.KNO3 C.Glucose D.Ca(OH)2 4) What is the pOH of the solution formed by mixing 15 mL of 0.10 M HNO₃ and 10 mL 0.15 M Ca(OH)₂? Express answer in 2 decimal places. 5) Which would produce a p-type semiconductor when mixed with silicon? Choose all possible correct answers. Select one or more: A.Gallium B.Lead C.Sulfur D.Boron E.Arsenic 6) The properties of two pure liquids, X and Y, are investigated. First, 50 uL each of X and Y were added on a glass slide. It was observed that X formed globules while Y spreaded throughout the slide. Furthermore, an instrument-based analysis showed that liquids X and Y have comparable molar masses. Based from the information above, which of these two: 1. Has a higher surface tension at a fixed temperature? Answer 2. Is more likely to have hydrogen bonding interactions among its molecules? Answer 3. Is more useful as a drying agent? Answer

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