Tag Archives: College Physics by Openstax

Problem 6-2: Conversion of units from rpm to revolutions per second and radians per second

Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?

Solution:

This is a problem on conversion of units. We are given a rotation in revolutions per minute and asked to convert this to revolutions per second and radians per second.

For the first part, we are asked to convert 6 rev/min to revolutions per second.

\begin{align*} \frac{6 \ \text{rev}}{\text{minute}} & = \frac{6 \ \text{rev}}{\bcancel{\text{minute}}} \times \frac{1 \ \bcancel{\text{minute}}}{60 \ \text{seconds}} \\ \\ & = 0.1 \ \text{rev/second} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

For the next part, we are going to convert 6 rev/min to radians per second.

\begin{align*} \frac{6 \ \text{rev}}{\text{minute}} & = \frac{6 \ \bcancel{\text{rev}}}{\bcancel{\text{minute}}} \times \frac{2\pi \ \text{radians}}{1 \ \bcancel{\text{rev}}} \times \frac{1 \ \bcancel{\text{minute}}}{60 \ \text{seconds}} \\ \\ & = 0.6283 \ \text{rad/sec} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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College Physics by Openstax Chapter 4 Problem 8

What is the deceleration of the rocket sled if it comes to rest in 1.1 s from a speed of 1000 km/h? (Such deceleration caused one test subject to black out and have temporary blindness.)

Solution:

We are given the following: $v_{0}=1000 \ \text{km/h}$ , $v_{f}=0 \ \text{km/h}$ , $\Delta t = 1.1 \ \text{s}$ .

The acceleration is computed as the change in velocity divided by the change in time.

\begin{align*} a & = \frac{\Delta v}{\Delta t} \\ a & = \frac{v_{f}-v_{o}}{\Delta t} \\ a & = \frac{\left( 0\ \text{km/h}-1000 \ \text{km/h} \right)\left( \frac{1000 \ \text{m}}{1\ \text{km}} \right) \left( \frac{1\ \text{h}}{3600\ \text{s}} \right)}{1.1\ \text{s}} \\ a & = -252.5\ \text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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College Physics by Openstax Chapter 4 Problem 7

(a) If the rocket sled shown in Figure 4.31 starts with only one rocket burning, what is the magnitude of its acceleration? Assume that the mass of the system is 2100 kg, the thrust $T$ is $2.4 \times 10^{4}$ N, and the force of friction opposing the motion is known to be 650 N. (b) Why is the acceleration not one-fourth of what it is with all rockets burning?

Solution:

Considering the direction of motion as the positive direction, we are given the following: $T=2.4 \times 10^4 \ \text{N}$ , $f=-650 \ \text{N}$ , and mass, $m=2100 \ \text{kg}$ .

Part A. The magnitude of the acceleration can be computed using Newton’s Second Law of Motion.

\begin{align*} \Sigma F & =ma \\ 2.4\times 10^4 \ \text{N}-650 \ \text{N} & = 2100 \ \text{kg}\times a \\ 23350 & = 2100 a \\ \frac{23350}{2100} & = \frac{\cancel{2100} a}{\cancel{2100}} \\ a & = \frac{23350}{2100} \\ a & = 11 \ \text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B. The acceleration is not one-fourth of what it was with all rockets burning because the frictional force is still as large as it was with all rockets burning. $\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)$

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College Physics by Openstax Chapter 4 Problem 6

The same rocket sled drawn in Figure 4.30 is decelerated at a rate of 196 m/s². What force is necessary to produce this deceleration? Assume that the rockets are off. The mass of the system is 2100 kg.

Solution:

Since the rockets are off, the only force acting on the sled is the friction $f$ . This force is against the direction of motion. By using Newton’s Second Law of Motion, we have.

\begin{align*} \Sigma F & = ma \\ -f & = ma \\ -f & = \left( 2100 \ \text{kg} \right)\left( -196 \ \text{m/s}^{2} \right) \\ -f & = -411600 \ \text{N} \\ f & = 411600 \ \text{N} \\ f & = 411.6 \ \text{kN} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

The force necessary to produce the given deceleration is 411.6 kN.

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College Physics by Openstax Chapter 4 Problem 5

In Figure 4.7, the net external force on the 24-kg mower is stated to be 51 N. If the force of friction opposing the motion is 24 N, what force F (in newtons) is the person exerting on the mower? Suppose the mower is moving at 1.5 m/s when the force F is removed. How far will the mower go before stopping?

Solution:

We can isolate the mower and expose the forces acting on it. This is the free-body diagram.

The free-body diagram of the mower: the force F exerted by the person and the friction force f exerted by the ground on the mower.

There are two forces acting on the mower in the horizontal directions:

$\textbf{F}:$ This is the force exerted by the person on the mower, and it is going to the right. This is the first unknown in the problem. We treat this as a positive force since it is directed to the right. The value of this force is $F=51 \ \text{N}$
$\textbf{f}:$ This is the friction force directed opposite the motion of the mower. We treat this as a negative force because it is directed to the left. The value of this force is $f=24 \ \text{N}$ .

Part A. The third force in the figure is the net force, $F_{net}$ . This is the vector sum of the forces $F$ and $f$ . That is

\begin{align*} F_{net} & =F-f \\ 51 \ \text{N} & = F-24 \ \text{N} \\ F & = 51 \ \text{N} +24 \ \text{N} \\ F & = 75 \ \text{N} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

The person should exert a force of 75 N to produce a net force of 51 N.

Part B. When the force F is removed, the friction is now the only force acting on the mower. The friction is acting opposite the direction of motion. The direction of motion is indicated by the blue arrow in the figure below, and the friction force is the red arrow.

The free-body diagram of the mower when force F is removed.

Using Newton’s Second Law, we can solve for the deceleration of the mower.

\begin{align*} \Sigma F & = ma \\ -24 \ \text{N} & = \left( 24 \ \text{kg} \right) \ a \\ \frac{-24 \ \text{N}}{24 \ \text{kg}} & = \frac{\cancel{24 \ \text{kg}}\ \ a}{\cancel{24 \ \text{kg}}} \\ a & = \frac{-24 \ \text{N}}{24 \ \text{kg}} \\ a & = -1 \ \text{m/s}^{2} \\ \end{align*}

Using this deceleration computed above, we can solve for the distance traveled by the mower before coming to stop.

\begin{align*} \left( v_{f} \right)^{2} & = \left( v_{o} \right)^{2}+2a\Delta x \\ \left( 0 \ \text{m/s} \right)^{2} & = \left( 1.5 \ \text{m/s} \right)^{2}+2\left( -1 \ \text{m/s}^{2} \right)\Delta x \\ 0 & = 2.25-2 \Delta x \\ 2 \Delta x & = 2.25 \\ \frac{\cancel{2}\Delta x}{\cancel{2}} & =\frac{2.25}{2} \\ \Delta x & = 1.125 \ \text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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College Physics by Openstax Chapter 4 Problem 4

Since astronauts in orbit are apparently weightless, a clever method of measuring their masses is needed to monitor their mass gains or losses to adjust diets. One way to do this is to exert a known force on an astronaut and measure the acceleration produced. Suppose a net external force of 50.0 N is exerted and the astronaut’s acceleration is measured to be 0.893 m/s². (a) Calculate her mass. (b) By exerting a force on the astronaut, the vehicle in which they orbit experiences an equal and opposite force. Discuss how this would affect the measurement of the astronaut’s acceleration. Propose a method in which recoil of the vehicle is avoided.

Solution:

We are given the following: $\sum F = 50.0 \ \text{N}$ , and $a=0.893 \ \text{m/s}^{2}$ .

Part A. We can solve for the mass, $m$ by using Newton’s second law of motion.

\begin{align*} \sum F & = ma \\ 50.0 \ \text{N} & = m \left( 0.893 \ \text{m/s}^{2} \right) \\ m & = \frac{50.0 \ \text{N}}{0.893 \ \text{m/s}^{2}} \\ m & = 56.0 \ \text{kg}\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B. The measured acceleration is equal to the sum of the accelerations of the astronauts and the ship. That is

a_{measured}=a_{astronaut}+a_{ship}

If a force acting on the astronaut came from something other than the spaceship, the spaceship would not undergo a recoil. $\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)$

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College Physics by Openstax Chapter 4 Problem 3

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate its acceleration.

Solution:

From Newton’s Second Law of Motion, $\sum F =ma$ . Substituting the given values, we have

\begin{align*} \sum F & = ma \\ 60.0 \ \text{N} & = \left( 4.50 \ \text{kg} \right) \ a \\ a & = \frac{60.0 \ \text{N}}{4.50 \ \text{kg}} \\ a & = 13.3 \ \text{m/s}^{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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College Physics by Openstax Chapter 4 Problem 2

If the sprinter from the previous problem accelerates at that rate for 20 m, and then maintains that velocity for the remainder of the 100-m dash, what will be his time for the race?

Solution:

Solving for the time it takes to reach the first 20 meters.

\begin{align*} \Delta x & =v_{0}t+\frac{1}{2}at^{2} \\ 20 \ \text{m} & = \left( 0 \ \text{m/s} \right)t + \frac{1}{2}\left( 4.20 \ \text{m/s}^{2} \right)t^{2} \\ 20 & = 2.1 t^{2} \\ \frac{20}{2.1}& = \frac{\cancel{2.1} \ t^{2}}{\cancel{2.1}} \\ t^{2} & = 9.5238 \\ \sqrt{t^{2}} & = \sqrt{9.5238} \\ t_{1} & = 3.09 \ \text{s} \end{align*}

We can compute the velocity of the sprinter at the end of the first 20 meters.

\begin{align*} v^2 & = v_{0}^2 + 2ax \\ v^2 & = \left( 0 \ \text{m/s} \right)^2 + 2\left( 4.20 \ \text{m/s}^2 \right) \left( 20 \ \text{m} \right) \\ v & = \sqrt{2\left( 4.20 \right)\left( 20 \right)} \ \text{m/s} \\ v & = 12.96 \ \text{m/s} \end{align*}

For the remaining 80 meters, the sprinter has a constant velocity of 12.96 m/s. The sprinter’s time to run the last 80 meters can be computed as follows.

\begin{align*} \Delta x & = vt \\ 80 \ \text{m} & = \left( 12.96 \ \text{m/s} \right)\ t_{2} \\ \frac{80 \ \text{m}}{12.96 \ \text{m/s}} & =\frac{\cancel{12.96 \ \text{m/s} } \ \ t_{2}}{\cancel{12.96 \ \text{m/s}}} \\ t_{2} & = \frac{80}{12.96} \ \text{s} \\ t_{2} & = 6.17 \ \text{s} \\ \end{align*}

The sprinter’s total time, $t_{T}$ , to finish the 100-m race is the sum of the two times.

\begin{align*} t_{T} & = t_{1} + t_{2} \\ t_{T} & = 3.09 \ \text{s} + 6.17 \ \text{s} \\ t_{T} & = 9.26 \ \text{s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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College Physics by Openstax Chapter 3 Problem 22

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in Figure 3.60, and then correctly calculates the length and orientation of the fourth side D. What is his result?

Solution:

For the four-sided plot to be closed, the resultant displacement of the four sides should be zero. The sum of the horizontal components should be zero, and the sum of the vertical components should also be equal to zero.

We need to solve for the components of each vector. Take into consideration that rightward and upward components are positive, while the reverse is negative.

For vector A, the components are

\begin{align*} A_x & = \left( 4.70 \ \text{km} \right) \cos 7.5^\circ \\ A_x & = 4.6598 \ \text{km} \end{align*}

\begin{align*} A_y & = -\left( 4.70 \ \text{km} \right) \sin 7.5^\circ \\ A_y & = -0.6135 \ \text{km} \end{align*}

The components of vector B are

\begin{align*} B_x & =-\left( 2.48 \ \text{km} \right) \sin 16^\circ \\ B_x & = -0.6836 \ \text{km} \end{align*}

\begin{align*} B_y & =\left( 2.48 \ \text{km} \right) \cos 16^\circ \\ B_y & =2.3839 \ \text{km} \end{align*}

For vector C, the components are

\begin{align*} C_x & = -\left( 3.02 \ \text{km} \right) \cos 19^\circ \\ C_x & = -2.8555 \ \text{km} \end{align*}

\begin{align*} C_y & = \left( 3.02 \ \text{km} \right) \sin 19^\circ \\ C_y & = 0.9832 \ \text{km} \end{align*}

Now, we need to take the sum of the x-components and equate it to zero. The x-component of D is unknown.

\begin{align*} A_x+B_x+C_x+D_x & =0 \\ 4.6598 \ \text{km}-0.6836 \ \text{km}-2.8555 \ \text{km}+ D_x & =0 \\ 1.1207 \ \text{km} +D_x & =0 \\ D_x & = -1.1207 \ \text{km} \end{align*}

We also need to take the sum of the y-component and equate it to zero to solve for the y-component of D.

\begin{align*} A_y +B_y+C_y+D_y & =0 \\ -0.6135 \ \text{km}+2.3839 \ \text{km}+0.9832 \ \text{km}+ D_y & =0 \\ 2.7536 \ \text{km} +D_y & =0 \\ D_y & = -2.7536 \ \text{km} \end{align*}

To solve for the distance of D, we shall use the Pythagorean Theorem.

\begin{align*} D & = \sqrt{\left( D_x \right)^2+\left( D_y \right)^2} \\ D & = \sqrt{\left( -1.1207 \ \text{km} \right)^2+\left( -2.7536 \ \text{km} \right)^2} \\ D & = 2.97 \ \text{km} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Then we can solve for θ using the tangent function. Since it is taken from the vertical axis, it can be solved by:

\begin{align*} \theta & = \tan^{-1} \left| \frac{D_x}{D_y} \right| \\ \theta & = \tan^{-1} \left| \frac{-1.1207 \ \text{km}}{-2.7536 \ \text{km}} \right| \\ \theta & = 22.1 ^ \circ \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

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College Physics by Openstax Chapter 3 Problem 21

You fly 32.0 km in a straight line in still air in the direction 35.0º south of west. (a) Find the distances you would have to fly straight south and then straight west to arrive at the same point. (This determination is equivalent to finding the components of the displacement along the south and west directions.) (b) Find the distances you would have to fly first in a direction 45.0º south of west and then in a direction 45.0º west of north. These are the components of the displacement along a different set of axes—one rotated 45º.

Solution:

Part A

Consider the illustration shown.

The south and west components of the 32.0 km distance are denoted by D_S and D_W, respectively. The values of these components are solved below:

\begin{align*} D_S & = \left( 32.0\ \text{km} \right) \sin 35.0 ^\circ \\ D_S & = 18.4^\circ \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

\begin{align*} D_W & = \left( 32.0\ \text{km} \right) \cos 35.0 ^\circ \\ D_W & = 26.2^\circ \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Part B

Consider the new set of axes (X-Y) as shown below. This new set of axes is rotated 45° from the original axes. Thus, axis X is 45° south of west, and axis Y is 45° west of north. First, we can obviously see that θ has a value of 10°.

Therefore, the components of the 32.0 km distance along X and Y axes are:

\begin{align*} D_X & = \left( 32.0 \ \text{km} \right) \cos 10^\circ \\ D_X & = 31.5^\circ \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

\begin{align*} D_Y & = \left( 32.0 \ \text{km} \right) \sin 10^\circ \\ D_Y & = 5.56^\circ \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

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Tag Archives: College Physics by Openstax

Problem 6-2: Conversion of units from rpm to revolutions per second and radians per second

Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?

Solution:

College Physics by Openstax Chapter 4 Problem 8

What is the deceleration of the rocket sled if it comes to rest in 1.1 s from a speed of 1000 km/h? (Such deceleration caused one test subject to black out and have temporary blindness.)

College Physics by Openstax Chapter 4 Problem 7

College Physics by Openstax Chapter 4 Problem 6

The same rocket sled drawn in Figure 4.30 is decelerated at a rate of 196 m/s². What force is necessary to produce this deceleration? Assume that the rockets are off. The mass of the system is 2100 kg.

College Physics by Openstax Chapter 4 Problem 5

College Physics by Openstax Chapter 4 Problem 4

College Physics by Openstax Chapter 4 Problem 3

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate its acceleration.

College Physics by Openstax Chapter 4 Problem 2

If the sprinter from the previous problem accelerates at that rate for 20 m, and then maintains that velocity for the remainder of the 100-m dash, what will be his time for the race?

College Physics by Openstax Chapter 3 Problem 22

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in Figure 3.60, and then correctly calculates the length and orientation of the fourth side D. What is his result?

College Physics by Openstax Chapter 3 Problem 21

Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?

Solution:

What is the deceleration of the rocket sled if it comes to rest in 1.1 s from a speed of 1000 km/h? (Such deceleration caused one test subject to black out and have temporary blindness.)

The same rocket sled drawn in Figure 4.30 is decelerated at a rate of 196 m/s2. What force is necessary to produce this deceleration? Assume that the rockets are off. The mass of the system is 2100 kg.

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate its acceleration.

If the sprinter from the previous problem accelerates at that rate for 20 m, and then maintains that velocity for the remainder of the 100-m dash, what will be his time for the race?

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in Figure 3.60, and then correctly calculates the length and orientation of the fourth side D. What is his result?

The same rocket sled drawn in Figure 4.30 is decelerated at a rate of 196 m/s². What force is necessary to produce this deceleration? Assume that the rockets are off. The mass of the system is 2100 kg.