Problem 5.6 Draw the Free-body Diagram of the Bar

5 Newton'south Laws of Movement

v.seven Drawing Gratuitous-Body Diagrams

Learning Objectives

By the end of the department, you will exist able to:

• Explain the rules for cartoon a costless-body diagram
• Construct free-trunk diagrams for unlike situations

The first step in describing and analyzing near phenomena in physics involves the careful drawing of a free-body diagram. Free-torso diagrams have been used in examples throughout this chapter. Remember that a gratuitous-body diagram must only include the external forces interim on the body of interest. Once we have drawn an accurate free-body diagram, we tin can apply Newton'southward first law if the body is in equilibrium (balanced forces; that is, [latex]{F}_{\text{net}}=0[/latex]) or Newton's second law if the body is accelerating (unbalanced force; that is, [latex]{F}_{\text{net}}\ne 0[/latex]).

In Forces, nosotros gave a cursory problem-solving strategy to assist you sympathize free-body diagrams. Here, we add some details to the strategy that will assistance you lot in constructing these diagrams.

Problem-Solving Strategy: Constructing Gratis-Torso Diagrams

Find the following rules when constructing a free-body diagram:

1. Draw the object under consideration; it does not accept to exist artistic. At kickoff, y'all may want to depict a circle effectually the object of involvement to be sure you focus on labeling the forces interim on the object. If you are treating the object as a particle (no size or shape and no rotation), represent the object equally a betoken. We often place this indicate at the origin of an xy-coordinate system.
2. Include all forces that act on the object, representing these forces as vectors. Consider the types of forces described in Common Forces—normal strength, friction, tension, and spring force—too as weight and applied force. Practice non include the net force on the object. With the exception of gravity, all of the forces we have discussed require direct contact with the object. Nonetheless, forces that the object exerts on its surroundings must not exist included. Nosotros never include both forces of an activeness-reaction pair.
3. Convert the free-body diagram into a more detailed diagram showing the x– and y-components of a given force (this is often helpful when solving a trouble using Newton's first or second law). In this case, place a squiggly line through the original vector to show that it is no longer in play—it has been replaced by its x– and y-components.
4. If there are ii or more objects, or bodies, in the problem, draw a separate free-body diagram for each object.

Note: If there is acceleration, we exercise not directly include it in the free-body diagram; however, it may help to point acceleration outside the free-torso diagram. You lot can label information technology in a different color to indicate that information technology is separate from the free-body diagram.

Permit's apply the problem-solving strategy in drawing a gratis-body diagram for a sled. In Figure(a), a sled is pulled past force P at an bending of [latex]xxx^\circ[/latex]. In part (b), we evidence a free-trunk diagram for this situation, as described past steps 1 and 2 of the problem-solving strategy. In part (c), we evidence all forces in terms of their x– and y-components, in keeping with step iii.

Figure 5.31 (a) A moving sled is shown as (b) a gratuitous-body diagram and (c) a gratuitous-body diagram with force components.

Instance

Two Blocks on an Inclined Plane

Construct the gratis-body diagram for object A and object B in Figure.

Strategy

We follow the four steps listed in the trouble-solving strategy.

Solution

Nosotros start by creating a diagram for the start object of interest. In Effigy(a), object A is isolated (circled) and represented by a dot.

Figure five.32 (a) The free-body diagram for isolated object A. (b) The costless-torso diagram for isolated object B. Comparing the two drawings, we see that friction acts in the opposite direction in the 2 figures. Because object A experiences a force that tends to pull it to the right, friction must human action to the left. Because object B experiences a component of its weight that pulls it to the left, downward the incline, the friction force must oppose it and human activity upwards the ramp. Friction always acts opposite the intended management of movement.

We now include whatever force that acts on the body. Here, no applied force is present. The weight of the object acts as a force pointing vertically downward, and the presence of the cord indicates a force of tension pointing abroad from the object. Object A has one interface and hence experiences a normal forcefulness, directed away from the interface. The source of this force is object B, and this normal strength is labeled accordingly. Since object B has a tendency to slide downwardly, object A has a tendency to slide upwardly with respect to the interface, and then the friction [latex]{f}_{\text{BA}}[/latex] is directed downwards parallel to the inclined plane.

As noted in step 4 of the problem-solving strategy, we then construct the costless-torso diagram in Effigy(b) using the same arroyo. Object B experiences ii normal forces and ii friction forces due to the presence of two contact surfaces. The interface with the inclined plane exerts external forces of [latex]{Due north}_{\text{B}}[/latex] and [latex]{f}_{\text{B}}[/latex], and the interface with object B exerts the normal force [latex]{N}_{\text{AB}}[/latex] and friction [latex]{f}_{\text{AB}}[/latex]; [latex]{Due north}_{\text{AB}}[/latex] is directed away from object B, and [latex]{f}_{\text{AB}}[/latex] is opposing the tendency of the relative motion of object B with respect to object A.

Significance

The object nether consideration in each part of this problem was circled in gray. When y'all are kickoff learning how to depict free-torso diagrams, y'all will find it helpful to circle the object before deciding what forces are acting on that detail object. This focuses your attention, preventing y'all from considering forces that are not interim on the trunk.

Case

Two Blocks in Contact

A strength is practical to 2 blocks in contact, as shown.

Strategy

Draw a free-torso diagram for each block. Be certain to consider Newton's tertiary police at the interface where the two blocks touch.

Solution

Significance[latex]{\mathbf{\overset{\to }{A}}}_{21}[/latex] is the activeness force of cake 2 on cake 1. [latex]{\mathbf{\overset{\to }{A}}}_{12}[/latex] is the reaction force of block 1 on cake 2. Nosotros employ these free-body diagrams in Applications of Newton's Laws.

Instance

Cake on the Tabular array (Coupled Blocks)

A block rests on the table, as shown. A light rope is attached to it and runs over a pulley. The other finish of the rope is attached to a 2d block. The two blocks are said to be coupled. Block [latex]{one thousand}_{ii}[/latex] exerts a force due to its weight, which causes the system (two blocks and a cord) to accelerate.

Strategy

We assume that the string has no mass so that we do not have to consider information technology every bit a separate object. Describe a free-body diagram for each block.

Solution

Significance

Each block accelerates (notice the labels shown for [latex]{\mathbf{\overset{\to }{a}}}_{1}[/latex] and [latex]{\mathbf{\overset{\to }{a}}}_{two}[/latex]); nevertheless, assuming the string remains taut, they accelerate at the same rate. Thus, we have [latex]{\mathbf{\overset{\to }{a}}}_{1}={\mathbf{\overset{\to }{a}}}_{ii}[/latex]. If we were to continue solving the problem, we could simply call the acceleration [latex]\mathbf{\overset{\to }{a}}[/latex]. Likewise, nosotros utilize two gratis-trunk diagrams because we are ordinarily finding tension T, which may crave us to apply a arrangement of ii equations in this type of problem. The tension is the aforementioned on both [latex]{thou}_{one}\,\text{and}\,{1000}_{2}[/latex].

Check Your Agreement

(a) Describe the free-torso diagram for the situation shown. (b) Redraw it showing components; employ x-axes parallel to the two ramps.

Prove Solution

Figure a shows a gratis trunk diagram of an object on a line that slopes down to the correct. Arrow T from the object points right and up, parallel to the gradient. Arrow N1 points left and upwards, perpendicular to the gradient. Pointer w1 points vertically down. Pointer w1x points left and down, parallel to the slope. Arrow w1y points right and down, perpendicular to the slope. Effigy b shows a free body diagram of an object on a line that slopes downwardly to the left. Arrow N2 from the object points right and upwardly, perpendicular to the slope. Arrow T points left and up, parallel to the slope. Arrow w2 points vertically down. Arrow w2y points left and down, perpendicular to the slope. Arrow w2x points correct and down, parallel to the slope.

View this simulation to predict, qualitatively, how an external force volition affect the speed and direction of an object's motion. Explain the effects with the help of a free-body diagram. Utilize costless-trunk diagrams to describe position, velocity, dispatch, and forcefulness graphs, and vice versa. Explain how the graphs chronicle to i another. Given a scenario or a graph, sketch all four graphs.

Summary

• To draw a free-body diagram, we draw the object of interest, draw all forces acting on that object, and resolve all force vectors into 10– and y-components. We must draw a separate free-body diagram for each object in the problem.
• A free-trunk diagram is a useful means of describing and analyzing all the forces that deed on a torso to determine equilibrium according to Newton's first constabulary or acceleration according to Newton's second constabulary.

Fundamental Equations

Net external force	[latex]{\mathbf{\overset{\to }{F}}}_{\text{net}}=\sum \mathbf{\overset{\to }{F}}={\mathbf{\overset{\to }{F}}}_{1}+{\mathbf{\overset{\to }{F}}}_{2}+\cdots[/latex]
Newton'south starting time police force	[latex]\mathbf{\overset{\to }{v}}=\,\text{constant when}\,{\mathbf{\overset{\to }{F}}}_{\text{net}}=\mathbf{\overset{\to }{0}}\,\text{North}[/latex]
Newton'due south second law, vector course	[latex]{\mathbf{\overset{\to }{F}}}_{\text{net}}=\sum \mathbf{\overset{\to }{F}}=chiliad\mathbf{\overset{\to }{a}}[/latex]
Newton'southward 2nd police force, scalar form	[latex]{F}_{\text{net}}=ma[/latex]
Newton's second law, component form	[latex]\sum {\mathbf{\overset{\to }{F}}}_{10}=m{\mathbf{\overset{\to }{a}}}_{ten}\text{,}\,\sum {\mathbf{\overset{\to }{F}}}_{y}=chiliad{\mathbf{\overset{\to }{a}}}_{y},\,\text{and}\,\sum {\mathbf{\overset{\to }{F}}}_{z}=1000{\mathbf{\overset{\to }{a}}}_{z}.[/latex]
Newton'due south second law, momentum form	[latex]{\mathbf{\overset{\to }{F}}}_{\text{net}}=\frac{d\mathbf{\overset{\to }{p}}}{dt}[/latex]
Definition of weight, vector course	[latex]\mathbf{\overset{\to }{west}}=g\mathbf{\overset{\to }{g}}[/latex]
Definition of weight, scalar course	[latex]w=mg[/latex]
Newton'southward third law	[latex]{\mathbf{\overset{\to }{F}}}_{\text{AB}}=\text{−}{\mathbf{\overset{\to }{F}}}_{\text{BA}}[/latex]
Normal strength on an object resting on a horizontal surface, vector form	[latex]\mathbf{\overset{\to }{North}}=\text{−}thou\mathbf{\overset{\to }{one thousand}}[/latex]
Normal force on an object resting on a horizontal surface, scalar form	[latex]North=mg[/latex]
Normal force on an object resting on an inclined plane, scalar form	[latex]N=mg\text{cos}\,\theta[/latex]
Tension in a cable supporting an object of mass m at rest, scalar form	[latex]T=westward=mg[/latex]

Conceptual Questions

In completing the solution for a problem involving forces, what do we do afterward constructing the free-body diagram? That is, what do we apply?

If a book is located on a table, how many forces should be shown in a gratis-body diagram of the book? Depict them.

Show Solution

Ii forces of different types: weight interim down and normal force acting upward

If the book in the previous question is in complimentary autumn, how many forces should be shown in a free-trunk diagram of the volume? Depict them.

Problems

A ball of mass k hangs at rest, suspended past a cord. (a) Sketch all forces. (b) Draw the free-body diagram for the ball.

A automobile moves forth a horizontal road. Describe a gratuitous-torso diagram; be sure to include the friction of the road that opposes the forwards motion of the car.

Evidence Solution

A runner pushes against the rails, every bit shown. (a) Provide a free-body diagram showing all the forces on the runner. (Hint: Place all forces at the middle of his body, and include his weight.) (b) Give a revised diagram showing the xy-component course.

The traffic low-cal hangs from the cables as shown. Depict a gratis-torso diagram on a coordinate plane for this situation.

Testify Solution

Boosted Issues

Two small forces, [latex]{\mathbf{\overset{\to }{F}}}_{1}=-ii.xl\mathbf{\hat{i}}-vi.10t\mathbf{\hat{j}}[/latex] N and [latex]{\mathbf{\overset{\to }{F}}}_{2}=eight.50\mathbf{\chapeau{i}}-9.70\mathbf{\hat{j}}[/latex] Northward, are exerted on a rogue asteroid by a pair of space tractors. (a) Detect the net force. (b) What are the magnitude and direction of the internet force? (c) If the mass of the asteroid is 125 kg, what acceleration does it experience (in vector form)? (d) What are the magnitude and direction of the acceleration?

Two forces of 25 and 45 N act on an object. Their directions differ by [latex]70^\circ[/latex]. The resulting acceleration has magnitude of [latex]x.0\,{\text{m/south}}^{two}.[/latex] What is the mass of the body?

A strength of 1600 N acts parallel to a ramp to push a 300-kg piano into a moving van. The ramp is inclined at [latex]20^\circ[/latex]. (a) What is the dispatch of the piano up the ramp? (b) What is the velocity of the piano when it reaches the top if the ramp is 4.0 grand long and the piano starts from rest?

Draw a gratuitous-body diagram of a diver who has entered the h2o, moved downwardly, and is acted on by an up force due to the water which balances the weight (that is, the diver is suspended).

Show Solution

For a swimmer who has merely jumped off a diving board, assume air resistance is negligible. The swimmer has a mass of lxxx.0 kg and jumps off a board ten.0 m above the water. Three seconds after inbound the h2o, her downwardly motion is stopped. What average upward force did the h2o exert on her?

(a) Observe an equation to determine the magnitude of the internet force required to stop a car of mass chiliad, given that the initial speed of the machine is [latex]{v}_{0}[/latex] and the stopping altitude is x. (b) Detect the magnitude of the cyberspace force if the mass of the car is 1050 kg, the initial speed is xl.0 km/h, and the stopping distance is 25.0 grand.

Show Solution

a. [latex]{F}_{\text{cyberspace}}=\frac{yard({v}^{two}-{v}_{0}{}^{ii})}{2x}[/latex]; b. 2590 N

A sailboat has a mass of [latex]i.l\times {10}^{3}[/latex] kg and is acted on by a force of [latex]2.00\times {10}^{3}[/latex] N toward the east, while the wind acts behind the sails with a force of [latex]3.00\times {10}^{3}[/latex] N in a direction [latex]45^\circ[/latex] n of east. Find the magnitude and direction of the resulting dispatch.

Find the acceleration of the trunk of mass 10.0 kg shown below.

Evidence Respond

[latex]\begin{assortment}{cc} {\mathbf{\overset{\to }{F}}}_{\text{net}}=4.05\mathbf{\lid{i}}+12.0\mathbf{\chapeau{j}}\text{N}\hfill \\ {\mathbf{\overset{\to }{F}}}_{\text{internet}}=m\mathbf{\overset{\to }{a}}\Rightarrow \mathbf{\overset{\to }{a}}=0.405\mathbf{\hat{i}}+ane.20\mathbf{\hat{j}}\,{\text{m/due south}}^{2}\hfill \end{array}[/latex]

A body of mass 2.0 kg is moving forth the x-axis with a speed of three.0 grand/s at the instant represented below. (a) What is the acceleration of the torso? (b) What is the body'south velocity 10.0 s later on? (c) What is its displacement after ten.0 southward?

Strength [latex]{\mathbf{\overset{\to }{F}}}_{\text{B}}[/latex] has twice the magnitude of force [latex]{\mathbf{\overset{\to }{F}}}_{\text{A}}.[/latex] Notice the direction in which the particle accelerates in this figure.

Show Answer

[latex]\begin{assortment}{cc} {\mathbf{\overset{\to }{F}}}_{\text{net}}={\mathbf{\overset{\to }{F}}}_{\text{A}}+{\mathbf{\overset{\to }{F}}}_{\text{B}}\hfill \\ {\mathbf{\overset{\to }{F}}}_{\text{net}}=A\mathbf{\hat{i}}+(-1.41A\mathbf{\lid{i}}-1.41A\mathbf{\lid{j}})\hfill \\ {\mathbf{\overset{\to }{F}}}_{\text{net}}=A(-0.41\mathbf{\hat{i}}-1.41\mathbf{\hat{j}})\hfill \\ \theta =254^\circ\hfill \end{array}[/latex]

(We add together [latex]180^\circ[/latex], because the angle is in quadrant IV.)

Shown below is a body of mass ane.0 kg under the influence of the forces [latex]{\mathbf{\overset{\to }{F}}}_{A}[/latex], [latex]{\mathbf{\overset{\to }{F}}}_{B}[/latex], and [latex]k\mathbf{\overset{\to }{1000}}[/latex]. If the trunk accelerates to the left at [latex]20\,{\text{thousand/s}}^{two}[/latex], what are [latex]{\mathbf{\overset{\to }{F}}}_{A}[/latex] and [latex]{\mathbf{\overset{\to }{F}}}_{B}[/latex]?

A force acts on a car of mass m and then that the speed v of the machine increases with position x every bit [latex]v=grand{x}^{2}[/latex], where k is constant and all quantities are in SI units. Find the force acting on the machine every bit a role of position.

Bear witness Solution

[latex]F=2kmx[/latex]; First, take the derivative of the velocity function to obtain [latex]a=2kx[/latex]. And so apply Newton's second law [latex]F=ma=m(2kx)=2kmx[/latex].

A 7.0-N forcefulness parallel to an incline is applied to a 1.0-kg crate. The ramp is tilted at [latex]20^\circ[/latex] and is frictionless. (a) What is the dispatch of the crate? (b) If all other atmospheric condition are the same but the ramp has a friction strength of i.nine N, what is the acceleration?

Ii boxes, A and B, are at residuum. Box A is on level ground, while box B rests on an inclined aeroplane tilted at angle [latex]\theta[/latex] with the horizontal. (a) Write expressions for the normal force acting on each cake. (b) Compare the two forces; that is, tell which ane is larger or whether they are equal in magnitude. (c) If the angle of incline is [latex]10^\circ[/latex], which force is greater?

Prove Solution

a. For box A, [latex]{N}_{\text{A}}=mg[/latex] and [latex]{N}_{\text{B}}=mg\,\text{cos}\,\theta[/latex]; b. [latex]{Due north}_{\text{A}} \gt {N}_{\text{B}}[/latex] because for [latex]\theta \lt xc^\circ[/latex], [latex]\text{cos}\,\theta \lt 1[/latex]; c. [latex]{N}_{\text{A}} \gt {N}_{\text{B}}[/latex] when [latex]\theta =x^\circ[/latex]

A mass of 250.0 thou is suspended from a spring hanging vertically. The spring stretches 6.00 cm. How much will the spring stretch if the suspended mass is 530.0 g?

Every bit shown below, two identical springs, each with the jump constant 20 N/m, support a 15.0-N weight. (a) What is the tension in spring A? (b) What is the amount of stretch of spring A from the residuum position?

Show Solution

a. eight.66 North; b. 0.433 m

Shown below is a 30.0-kg block resting on a frictionless ramp inclined at [latex]60^\circ[/latex] to the horizontal. The block is held by a spring that is stretched 5.0 cm. What is the strength abiding of the spring?

In building a house, carpenters use nails from a large box. The box is suspended from a spring twice during the mean solar day to measure the usage of nails. At the beginning of the day, the leap stretches 50 cm. At the terminate of the twenty-four hour period, the spring stretches 30 cm. What fraction or percentage of the nails have been used?

Show Solution

0.forty or forty%

A force is applied to a block to move it up a [latex]30^\circ[/latex] incline. The incline is frictionless. If [latex]F=65.0\,\text{N}[/latex] and [latex]One thousand=5.00\,\text{kg}[/latex], what is the magnitude of the acceleration of the block?

Two forces are applied to a five.0-kg object, and it accelerates at a rate of [latex]2.0\,{\text{thousand/s}}^{two}[/latex] in the positive y-direction. If one of the forces acts in the positive ten-management with magnitude 12.0 Northward, detect the magnitude of the other forcefulness.

The block on the right shown beneath has more than mass than the block on the left ([latex]{thousand}_{2} \gt {yard}_{1}[/latex]). Describe free-torso diagrams for each cake.

Claiming Problems

If two tugboats pull on a disabled vessel, as shown here in an overhead view, the disabled vessel volition be pulled along the direction indicated by the effect of the exerted forces. (a) Draw a gratuitous-torso diagram for the vessel. Presume no friction or elevate forces affect the vessel. (b) Did you lot include all forces in the overhead view in your free-body diagram? Why or why non?

Show Solution

a.

b. No; [latex]{\mathbf{\overset{\to }{F}}}_{\text{R}}[/latex] is not shown, considering it would replace [latex]{\mathbf{\overset{\to }{F}}}_{1}[/latex] and [latex]{\mathbf{\overset{\to }{F}}}_{2}[/latex]. (If we want to prove it, we could describe it and then place squiggly lines on [latex]{\mathbf{\overset{\to }{F}}}_{one}[/latex] and [latex]{\mathbf{\overset{\to }{F}}}_{2}[/latex] to show that they are no longer considered.

A ten.0-kg object is initially moving eastward at 15.0 m/s. Then a force acts on information technology for 2.00 s, after which it moves northwest, besides at 15.0 1000/southward. What are the magnitude and direction of the average strength that acted on the object over the 2.00-due south interval?

On June 25, 1983, shot-doodle Udo Beyer of Eastward Deutschland threw the seven.26-kg shot 22.22 m, which at that fourth dimension was a globe record. (a) If the shot was released at a tiptop of 2.twenty m with a projection angle of [latex]45.0^\circ[/latex], what was its initial velocity? (b) If while in Beyer'southward hand the shot was accelerated uniformly over a altitude of 1.20 m, what was the cyberspace force on it?

Bear witness Solution

a. 14.1 m/southward; b. 601 N

A body of mass 1000 moves in a horizontal direction such that at time t its position is given past [latex]x(t)=a{t}^{four}+b{t}^{3}+ct,[/latex] where a, b, and c are constants. (a) What is the acceleration of the body? (b) What is the fourth dimension-dependent force acting on the torso?

A body of mass m has initial velocity [latex]{v}_{0}[/latex] in the positive x-direction. It is acted on by a constant force F for fourth dimension t until the velocity becomes zero; the forcefulness continues to human activity on the trunk until its velocity becomes [latex]\text{−}{v}_{0}[/latex] in the same amount of fourth dimension. Write an expression for the total distance the trunk travels in terms of the variables indicated.

Bear witness Solution

[latex]\frac{F}{one thousand}{t}^{ii}[/latex]

The velocities of a 3.0-kg object at [latex]t=6.0\,\text{s}[/latex] and [latex]t=eight.0\,\text{s}[/latex] are [latex](3.0\mathbf{\hat{i}}-6.0\mathbf{\chapeau{j}}+4.0\mathbf{\hat{k}})\,\text{m/s}[/latex] and [latex](-2.0\mathbf{\hat{i}}+four.0\mathbf{\hat{k}})\,\text{m/s}[/latex], respectively. If the object is moving at abiding dispatch, what is the strength interim on information technology?

A 120-kg astronaut is riding in a rocket sled that is sliding forth an inclined plane. The sled has a horizontal component of acceleration of [latex]5.0\,\text{thou}\text{/}{\text{due south}}^{2}[/latex] and a down component of [latex]3.8\,\text{grand}\text{/}{\text{s}}^{2}[/latex]. Calculate the magnitude of the force on the rider by the sled. (Hint: Call back that gravitational acceleration must be considered.)

Two forces are acting on a 5.0-kg object that moves with dispatch [latex]2.0\,{\text{m/s}}^{2}[/latex] in the positive y-direction. If 1 of the forces acts in the positive x-direction and has magnitude of 12 N, what is the magnitude of the other force?

Suppose that you are viewing a soccer game from a helicopter above the playing field. Two soccer players simultaneously boot a stationary soccer ball on the flat field; the soccer ball has mass 0.420 kg. The first actor kicks with forcefulness 162 N at [latex]ix.0^\circ[/latex] north of west. At the same instant, the 2nd player kicks with forcefulness 215 Northward at [latex]15^\circ[/latex] east of south. Detect the dispatch of the ball in [latex]\mathbf{\hat{i}}[/latex] and [latex]\mathbf{\hat{j}}[/latex] form.

Evidence Solution

[latex]\mathbf{\overset{\to }{a}}=-248\mathbf{\lid{i}}-433\mathbf{\chapeau{j}}\text{thou}\text{/}{\text{south}}^{2}[/latex]

A 10.0-kg mass hangs from a jump that has the spring constant 535 N/m. Notice the position of the finish of the spring away from its rest position. (Use [latex]m=9.80\,{\text{m/s}}^{2}[/latex].)

A 0.0502-kg pair of fuzzy dice is attached to the rearview mirror of a car past a brusque cord. The car accelerates at abiding rate, and the dice hang at an angle of [latex]3.xx^\circ[/latex] from the vertical because of the car'southward dispatch. What is the magnitude of the acceleration of the auto?

Bear witness Solution

[latex]0.548\,{\text{m/s}}^{ii}[/latex]

At a circus, a donkey pulls on a sled carrying a small-scale clown with a force given past [latex]two.48\mathbf{\hat{i}}+four.33\mathbf{\hat{j}}\,\text{N}[/latex]. A horse pulls on the aforementioned sled, aiding the hapless donkey, with a strength of [latex]6.56\mathbf{\hat{i}}+5.33\mathbf{\hat{j}}\,\text{N}[/latex]. The mass of the sled is 575 kg. Using [latex]\mathbf{\hat{i}}[/latex] and [latex]\mathbf{\hat{j}}[/latex] grade for the reply to each problem, discover (a) the net force on the sled when the two animals act together, (b) the acceleration of the sled, and (c) the velocity after 6.fifty s.

Hanging from the ceiling over a baby bed, well out of babe'due south achieve, is a cord with plastic shapes, as shown here. The string is taut (there is no slack), as shown by the straight segments. Each plastic shape has the same mass m, and they are equally spaced by a altitude d, as shown. The angles labeled [latex]\theta[/latex] depict the bending formed by the terminate of the string and the ceiling at each end. The center length of sting is horizontal. The remaining ii segments each form an angle with the horizontal, labeled [latex]\varphi[/latex]. Allow [latex]{T}_{1}[/latex] be the tension in the leftmost department of the cord, [latex]{T}_{2}[/latex] be the tension in the department adjacent to it, and [latex]{T}_{3}[/latex] be the tension in the horizontal segment. (a) Find an equation for the tension in each section of the string in terms of the variables m, g, and [latex]\theta[/latex]. (b) Find the angle [latex]\varphi[/latex] in terms of the angle [latex]\theta[/latex]. (c) If [latex]\theta =5.ten^\circ[/latex], what is the value of [latex]\varphi[/latex]? (d) Find the distance x between the endpoints in terms of d and [latex]\theta[/latex].

Show Solution

a. [latex]{T}_{1}=\frac{2mg}{\text{sin}\,\theta }[/latex], [latex]{T}_{2}=\frac{mg}{\text{sin}(\text{arctan}(\frac{1}{2}\text{tan}\,\theta ))}[/latex], [latex]{T}_{3}=\frac{2mg}{\text{tan}\,\theta };[/latex] b. [latex]\varphi =\text{arctan}(\frac{1}{2}\text{tan}\,\theta )[/latex]; c. [latex]2.56^\circ[/latex]; (d) [latex]x=d(2\,\text{cos}\,\theta +2\,\text{cos}(\text{arctan}(\frac{1}{2}\text{tan}\,\theta ))+i)[/latex]

A bullet shot from a rifle has mass of 10.0 g and travels to the right at 350 m/s. It strikes a target, a large bag of sand, penetrating information technology a distance of 34.0 cm. Discover the magnitude and direction of the retarding forcefulness that slows and stops the bullet.

An object is acted on past three simultaneous forces: [latex]{\mathbf{\overset{\to }{F}}}_{one}=(-3.00\mathbf{\lid{i}}+2.00\mathbf{\chapeau{j}})\,\text{N}[/latex], [latex]{\mathbf{\overset{\to }{F}}}_{2}=(6.00\mathbf{\hat{i}}-4.00\mathbf{\chapeau{j}})\,\text{N}[/latex], and [latex]{\mathbf{\overset{\to }{F}}}_{3}=(2.00\mathbf{\hat{i}}+5.00\mathbf{\lid{j}})\,\text{North}[/latex]. The object experiences dispatch of [latex]4.23\,{\text{1000/southward}}^{2}[/latex]. (a) Find the acceleration vector in terms of chiliad. (b) Observe the mass of the object. (c) If the object begins from rest, find its speed after 5.00 due south. (d) Observe the components of the velocity of the object after 5.00 s.

Bear witness Solution

a. [latex]\mathbf{\overset{\to }{a}}=(\frac{v.00}{m}\mathbf{\hat{i}}+\frac{3.00}{m}\mathbf{\lid{j}})\,\text{m}\text{/}{\text{s}}^{2};[/latex] b. one.38 kg; c. 21.two m/s; d. [latex]\mathbf{\overset{\to }{v}}=(18.1\mathbf{\hat{i}}+10.9\mathbf{\hat{j}})\,\text{m}\text{/}{\text{s}}^{two}[/latex]

In a particle accelerator, a proton has mass [latex]i.67\times {10}^{-27}\,\text{kg}[/latex] and an initial speed of [latex]2.00\times {x}^{5}\,\text{m}\text{/}\text{s.}[/latex] It moves in a directly line, and its speed increases to [latex]9.00\times {10}^{5}\,\text{thousand}\text{/}\text{s}[/latex] in a distance of 10.0 cm. Assume that the acceleration is constant. Discover the magnitude of the force exerted on the proton.

A drone is being directed across a frictionless ice-covered lake. The mass of the drone is i.l kg, and its velocity is [latex]3.00\mathbf{\hat{i}}\text{chiliad}\text{/}\text{due south}[/latex]. After 10.0 due south, the velocity is [latex]nine.00\mathbf{\hat{i}}+4.00\mathbf{\hat{j}}\text{thousand}\text{/}\text{southward}[/latex]. If a abiding force in the horizontal direction is causing this change in motion, find (a) the components of the strength and (b) the magnitude of the strength.

Show Solution

a. [latex]0.900\mathbf{\lid{i}}+0.600\mathbf{\hat{j}}\,\text{N}[/latex]; b. ane.08 North

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Source: https://pressbooks.online.ucf.edu/phy2048tjb/chapter/5-7-drawing-free-body-diagrams/

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