Rules for Drawing Free Body Diagrams

five Newton's Laws of Motility

v.7 Cartoon Free-Body Diagrams

Learning Objectives

By the end of the section, you will be able to:

• Explicate the rules for drawing a free-body diagram
• Construct complimentary-body diagrams for different situations

The kickoff stride in describing and analyzing most phenomena in physics involves the careful cartoon of a costless-trunk diagram. Free-body diagrams have been used in examples throughout this chapter. Call up that a gratuitous-body diagram must only include the external forces acting on the trunk of interest. Once we accept drawn an accurate free-body diagram, we can apply Newton'southward starting time constabulary if the body is in equilibrium (balanced forces; that is, [latex]{F}_{\text{net}}=0[/latex]) or Newton's 2nd law if the torso is accelerating (unbalanced force; that is, [latex]{F}_{\text{net}}\ne 0[/latex]).

In Forces, nosotros gave a brief trouble-solving strategy to help you understand free-body diagrams. Here, we add some details to the strategy that will assist you in constructing these diagrams.

Problem-Solving Strategy: Constructing Free-Torso Diagrams

Discover the following rules when constructing a complimentary-body diagram:

1. Depict the object under consideration; it does not have to be creative. At first, yous may want to describe a circumvolve around the object of interest to be certain you lot focus on labeling the forces interim on the object. If you are treating the object equally a particle (no size or shape and no rotation), represent the object as a point. We ofttimes identify this point 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 force, friction, tension, and leap forcefulness—as well as weight and practical forcefulness. Exercise non include the net force on the object. With the exception of gravity, all of the forces we accept discussed crave direct contact with the object. Withal, forces that the object exerts on its surroundings must non be included. We never include both forces of an action-reaction pair.
3. Convert the costless-trunk diagram into a more detailed diagram showing the 10– and y-components of a given force (this is often helpful when solving a problem using Newton'southward first or second police force). 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 two or more objects, or bodies, in the trouble, draw a separate costless-body diagram for each object.

Annotation: If there is dispatch, nosotros do non straight include information technology in the complimentary-trunk diagram; however, it may help to indicate acceleration outside the free-body diagram. You lot tin can label it in a dissimilar color to indicate that it is dissever from the complimentary-trunk diagram.

Permit's apply the trouble-solving strategy in drawing a free-body diagram for a sled. In Effigy(a), a sled is pulled by force P at an angle of [latex]xxx^\circ[/latex]. In part (b), we evidence a free-body diagram for this situation, equally described past steps 1 and 2 of the problem-solving strategy. In role (c), we show all forces in terms of their x– and y-components, in keeping with pace 3.

Figure v.31 (a) A moving sled is shown as (b) a gratuitous-body diagram and (c) a complimentary-torso diagram with force components.

Example

Two Blocks on an Inclined Plane

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

Strategy

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

Solution

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

Effigy 5.32 (a) The free-torso diagram for isolated object A. (b) The gratis-trunk diagram for isolated object B. Comparing the two drawings, nosotros run into that friction acts in the contrary direction in the ii figures. Because object A experiences a force that tends to pull it to the right, friction must deed to the left. Because object B experiences a component of its weight that pulls it to the left, downwardly the incline, the friction strength must oppose it and act upward the ramp. Friction always acts opposite the intended direction of motion.

We now include whatever force that acts on the torso. Hither, no applied forcefulness 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 away from the object. Object A has i interface and hence experiences a normal force, directed away from the interface. The source of this forcefulness is object B, and this normal strength is labeled appropriately. Since object B has a tendency to slide downwards, object A has a tendency to slide up with respect to the interface, and so the friction [latex]{f}_{\text{BA}}[/latex] is directed downwardly parallel to the inclined airplane.

As noted in step 4 of the trouble-solving strategy, we then construct the gratis-body diagram in Figure(b) using the aforementioned approach. Object B experiences two normal forces and 2 friction forces due to the presence of two contact surfaces. The interface with the inclined plane exerts external forces of [latex]{N}_{\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]{N}_{\text{AB}}[/latex] is directed away from object B, and [latex]{f}_{\text{AB}}[/latex] is opposing the trend 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 you are outset learning how to describe free-body diagrams, y'all will discover it helpful to circle the object earlier deciding what forces are acting on that particular object. This focuses your attention, preventing you from considering forces that are not acting on the trunk.

Example

Two Blocks in Contact

A forcefulness is applied to two blocks in contact, as shown.

Strategy

Draw a gratuitous-body diagram for each cake. Be certain to consider Newton's 3rd law at the interface where the two blocks touch.

Solution

Significance[latex]{\mathbf{\overset{\to }{A}}}_{21}[/latex] is the action force of cake 2 on block one. [latex]{\mathbf{\overset{\to }{A}}}_{12}[/latex] is the reaction strength of block 1 on block two. We use these costless-body diagrams in Applications of Newton's Laws.

Case

Cake on the Table (Coupled Blocks)

A block rests on the tabular array, equally shown. A light rope is fastened to it and runs over a pulley. The other end of the rope is fastened to a second block. The two blocks are said to exist coupled. Cake [latex]{m}_{2}[/latex] exerts a force due to its weight, which causes the arrangement (two blocks and a string) to accelerate.

Strategy

We assume that the string has no mass so that we practice non take to consider it as a separate object. Draw a free-body diagram for each cake.

Solution

Significance

Each cake accelerates (find the labels shown for [latex]{\mathbf{\overset{\to }{a}}}_{ane}[/latex] and [latex]{\mathbf{\overset{\to }{a}}}_{two}[/latex]); all the same, 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 nosotros were to go on solving the problem, nosotros could simply call the acceleration [latex]\mathbf{\overset{\to }{a}}[/latex]. Also, we use two gratuitous-trunk diagrams because we are unremarkably finding tension T, which may require usa to employ a system of two equations in this blazon of problem. The tension is the same on both [latex]{g}_{one}\,\text{and}\,{chiliad}_{ii}[/latex].

Check Your Understanding

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

Show Solution

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

View this simulation to predict, qualitatively, how an external force will affect the speed and direction of an object's motility. Explain the furnishings with the help of a gratis-body diagram. Utilize complimentary-body diagrams to depict position, velocity, dispatch, and force graphs, and vice versa. Explain how the graphs relate to one another. Given a scenario or a graph, sketch all four graphs.

Summary

• To describe a free-body diagram, we depict 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-torso diagram for each object in the problem.
• A free-body diagram is a useful means of describing and analyzing all the forces that act on a body to make up one's mind equilibrium co-ordinate to Newton's first constabulary or dispatch co-ordinate to Newton's 2d law.

Central Equations

Internet external force	[latex]{\mathbf{\overset{\to }{F}}}_{\text{cyberspace}}=\sum \mathbf{\overset{\to }{F}}={\mathbf{\overset{\to }{F}}}_{1}+{\mathbf{\overset{\to }{F}}}_{2}+\cdots[/latex]
Newton's first police	[latex]\mathbf{\overset{\to }{v}}=\,\text{constant when}\,{\mathbf{\overset{\to }{F}}}_{\text{net}}=\mathbf{\overset{\to }{0}}\,\text{North}[/latex]
Newton's second police, vector form	[latex]{\mathbf{\overset{\to }{F}}}_{\text{internet}}=\sum \mathbf{\overset{\to }{F}}=1000\mathbf{\overset{\to }{a}}[/latex]
Newton's second law, scalar class	[latex]{F}_{\text{net}}=ma[/latex]
Newton'south 2d police, component grade	[latex]\sum {\mathbf{\overset{\to }{F}}}_{x}=m{\mathbf{\overset{\to }{a}}}_{10}\text{,}\,\sum {\mathbf{\overset{\to }{F}}}_{y}=m{\mathbf{\overset{\to }{a}}}_{y},\,\text{and}\,\sum {\mathbf{\overset{\to }{F}}}_{z}=g{\mathbf{\overset{\to }{a}}}_{z}.[/latex]
Newton's second law, momentum grade	[latex]{\mathbf{\overset{\to }{F}}}_{\text{net}}=\frac{d\mathbf{\overset{\to }{p}}}{dt}[/latex]
Definition of weight, vector form	[latex]\mathbf{\overset{\to }{w}}=m\mathbf{\overset{\to }{g}}[/latex]
Definition of weight, scalar form	[latex]west=mg[/latex]
Newton's third law	[latex]{\mathbf{\overset{\to }{F}}}_{\text{AB}}=\text{−}{\mathbf{\overset{\to }{F}}}_{\text{BA}}[/latex]
Normal force on an object resting on a horizontal surface, vector form	[latex]\mathbf{\overset{\to }{Northward}}=\text{−}m\mathbf{\overset{\to }{m}}[/latex]
Normal strength on an object resting on a horizontal surface, scalar class	[latex]N=mg[/latex]
Normal strength on an object resting on an inclined aeroplane, scalar class	[latex]Northward=mg\text{cos}\,\theta[/latex]
Tension in a cable supporting an object of mass m at rest, scalar form	[latex]T=due west=mg[/latex]

Conceptual Questions

In completing the solution for a trouble involving forces, what do we practise after constructing the free-body diagram? That is, what practise nosotros utilize?

If a volume is located on a table, how many forces should exist shown in a free-body diagram of the volume? Depict them.

Show Solution

Two forces of unlike types: weight acting downward and normal forcefulness acting upwards

If the book in the previous question is in free fall, how many forces should be shown in a free-body diagram of the book? Draw them.

Problems

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

A car moves along a horizontal road. Draw a free-torso diagram; be sure to include the friction of the road that opposes the forrad movement of the auto.

Show Solution

A runner pushes against the track, as shown. (a) Provide a free-body diagram showing all the forces on the runner. (Hint: Place all forces at the heart of his body, and include his weight.) (b) Give a revised diagram showing the xy-component grade.

The traffic light hangs from the cables as shown. Draw a free-body diagram on a coordinate airplane for this situation.

Evidence Solution

Additional Problems

Two small forces, [latex]{\mathbf{\overset{\to }{F}}}_{ane}=-2.40\mathbf{\hat{i}}-6.10t\mathbf{\hat{j}}[/latex] N and [latex]{\mathbf{\overset{\to }{F}}}_{2}=8.50\mathbf{\hat{i}}-9.70\mathbf{\hat{j}}[/latex] N, are exerted on a rogue asteroid by a pair of space tractors. (a) Observe the net force. (b) What are the magnitude and management of the net strength? (c) If the mass of the asteroid is 125 kg, what dispatch does it experience (in vector course)? (d) What are the magnitude and direction of the dispatch?

Ii forces of 25 and 45 N human activity on an object. Their directions differ by [latex]lxx^\circ[/latex]. The resulting acceleration has magnitude of [latex]10.0\,{\text{thou/s}}^{ii}.[/latex] What is the mass of the body?

A force of 1600 Due north acts parallel to a ramp to button a 300-kg pianoforte into a moving van. The ramp is inclined at [latex]xx^\circ[/latex]. (a) What is the acceleration of the piano up the ramp? (b) What is the velocity of the piano when it reaches the elevation if the ramp is 4.0 thousand long and the pianoforte starts from rest?

Draw a gratis-body diagram of a diver who has entered the water, moved downward, and is acted on by an upward strength 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, presume air resistance is negligible. The swimmer has a mass of 80.0 kg and jumps off a board ten.0 m above the water. Three seconds afterwards inbound the water, her downwardly move is stopped. What boilerplate upwards force did the water exert on her?

(a) Discover an equation to determine the magnitude of the net force required to stop a machine of mass k, given that the initial speed of the auto is [latex]{v}_{0}[/latex] and the stopping distance is x. (b) Find the magnitude of the internet force if the mass of the car is 1050 kg, the initial speed is forty.0 km/h, and the stopping distance is 25.0 yard.

Show Solution

a. [latex]{F}_{\text{net}}=\frac{m({v}^{2}-{v}_{0}{}^{ii})}{2x}[/latex]; b. 2590 North

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

Detect the acceleration of the body of mass ten.0 kg shown below.

Show Answer

[latex]\begin{array}{cc} {\mathbf{\overset{\to }{F}}}_{\text{net}}=4.05\mathbf{\hat{i}}+12.0\mathbf{\chapeau{j}}\text{N}\hfill \\ {\mathbf{\overset{\to }{F}}}_{\text{internet}}=grand\mathbf{\overset{\to }{a}}\Rightarrow \mathbf{\overset{\to }{a}}=0.405\mathbf{\hat{i}}+one.20\mathbf{\lid{j}}\,{\text{1000/s}}^{2}\hfill \finish{array}[/latex]

A body of mass 2.0 kg is moving along the 10-axis with a speed of 3.0 one thousand/due south at the instant represented below. (a) What is the acceleration of the body? (b) What is the body's velocity 10.0 s later? (c) What is its deportation after 10.0 south?

Force [latex]{\mathbf{\overset{\to }{F}}}_{\text{B}}[/latex] has twice the magnitude of strength [latex]{\mathbf{\overset{\to }{F}}}_{\text{A}}.[/latex] Find the direction in which the particle accelerates in this effigy.

Show Respond

[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{\chapeau{i}}-1.41A\mathbf{\hat{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 [latex]180^\circ[/latex], because the angle is in quadrant Four.)

Shown below is a body of mass 1.0 kg under the influence of the forces [latex]{\mathbf{\overset{\to }{F}}}_{A}[/latex], [latex]{\mathbf{\overset{\to }{F}}}_{B}[/latex], and [latex]yard\mathbf{\overset{\to }{1000}}[/latex]. If the torso accelerates to the left at [latex]xx\,{\text{grand/due south}}^{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 so that the speed v of the automobile increases with position x equally [latex]v=k{x}^{2}[/latex], where m is constant and all quantities are in SI units. Find the force acting on the automobile as a function of position.

Prove Solution

[latex]F=2kmx[/latex]; Outset, accept the derivative of the velocity office to obtain [latex]a=2kx[/latex]. Then apply Newton's second law [latex]F=ma=1000(2kx)=2kmx[/latex].

A vii.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 acceleration of the crate? (b) If all other atmospheric condition are the same but the ramp has a friction force of 1.9 N, what is the acceleration?

2 boxes, A and B, are at residuum. Box A is on level ground, while box B rests on an inclined plane tilted at angle [latex]\theta[/latex] with the horizontal. (a) Write expressions for the normal force acting on each block. (b) Compare the 2 forces; that is, tell which one is larger or whether they are equal in magnitude. (c) If the angle of incline is [latex]10^\circ[/latex], which forcefulness 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]{North}_{\text{A}} \gt {Due north}_{\text{B}}[/latex] considering for [latex]\theta \lt ninety^\circ[/latex], [latex]\text{cos}\,\theta \lt 1[/latex]; c. [latex]{Due north}_{\text{A}} \gt {N}_{\text{B}}[/latex] when [latex]\theta =ten^\circ[/latex]

A mass of 250.0 k is suspended from a jump hanging vertically. The spring stretches half-dozen.00 cm. How much will the jump stretch if the suspended mass is 530.0 thousand?

As shown below, 2 identical springs, each with the spring constant 20 N/yard, support a xv.0-N weight. (a) What is the tension in spring A? (b) What is the amount of stretch of jump A from the rest position?

Show Solution

a. viii.66 Due north; b. 0.433 m

Shown below is a xxx.0-kg block resting on a frictionless ramp inclined at [latex]60^\circ[/latex] to the horizontal. The block is held by a bound that is stretched 5.0 cm. What is the force constant of the bound?

In building a business firm, carpenters apply nails from a big box. The box is suspended from a jump twice during the day to measure the usage of nails. At the kickoff of the day, the spring stretches 50 cm. At the stop of the day, the bound stretches 30 cm. What fraction or percentage of the nails have been used?

Testify Solution

0.40 or 40%

A force is applied to a cake to movement information technology up a [latex]30^\circ[/latex] incline. The incline is frictionless. If [latex]F=65.0\,\text{North}[/latex] and [latex]Chiliad=5.00\,\text{kg}[/latex], what is the magnitude of the acceleration of the block?

Two forces are applied to a 5.0-kg object, and information technology accelerates at a charge per unit of [latex]ii.0\,{\text{m/s}}^{2}[/latex] in the positive y-management. If one of the forces acts in the positive x-direction with magnitude 12.0 North, notice the magnitude of the other force.

The block on the right shown below has more mass than the block on the left ([latex]{m}_{2} \gt {thousand}_{1}[/latex]). Depict free-body diagrams for each block.

Challenge Problems

If two tugboats pull on a disabled vessel, as shown hither in an overhead view, the disabled vessel will be pulled along the management indicated by the result of the exerted forces. (a) Draw a costless-torso diagram for the vessel. Assume no friction or drag forces impact the vessel. (b) Did yous include all forces in the overhead view in your free-body diagram? Why or why not?

Show Solution

a.

b. No; [latex]{\mathbf{\overset{\to }{F}}}_{\text{R}}[/latex] is not shown, considering it would supplant [latex]{\mathbf{\overset{\to }{F}}}_{1}[/latex] and [latex]{\mathbf{\overset{\to }{F}}}_{2}[/latex]. (If nosotros desire to prove it, nosotros could draw it and and so identify squiggly lines on [latex]{\mathbf{\overset{\to }{F}}}_{1}[/latex] and [latex]{\mathbf{\overset{\to }{F}}}_{2}[/latex] to show that they are no longer considered.

A 10.0-kg object is initially moving east at fifteen.0 k/s. Then a force acts on it for 2.00 s, later on which it moves northwest, as well at 15.0 yard/south. What are the magnitude and direction of the boilerplate strength that acted on the object over the 2.00-s interval?

On June 25, 1983, shot-putter Udo Beyer of East Federal republic of germany threw the 7.26-kg shot 22.22 m, which at that fourth dimension was a world record. (a) If the shot was released at a height of 2.20 m with a projection bending of [latex]45.0^\circ[/latex], what was its initial velocity? (b) If while in Beyer'southward mitt the shot was accelerated uniformly over a distance of i.twenty m, what was the net force on it?

Show Solution

a. 14.i m/s; b. 601 N

A body of mass m moves in a horizontal management such that at time t its position is given by [latex]ten(t)=a{t}^{4}+b{t}^{three}+ct,[/latex] where a, b, and c are constants. (a) What is the acceleration of the body? (b) What is the fourth dimension-dependent strength acting on the trunk?

A torso of mass m has initial velocity [latex]{v}_{0}[/latex] in the positive ten-management. It is acted on by a abiding strength F for time t until the velocity becomes goose egg; the force continues to human action on the body until its velocity becomes [latex]\text{−}{v}_{0}[/latex] in the same amount of time. Write an expression for the full distance the body travels in terms of the variables indicated.

Show Solution

[latex]\frac{F}{m}{t}^{two}[/latex]

The velocities of a 3.0-kg object at [latex]t=half-dozen.0\,\text{due south}[/latex] and [latex]t=8.0\,\text{southward}[/latex] are [latex](three.0\mathbf{\hat{i}}-half-dozen.0\mathbf{\lid{j}}+4.0\mathbf{\hat{g}})\,\text{grand/s}[/latex] and [latex](-two.0\mathbf{\hat{i}}+4.0\mathbf{\hat{k}})\,\text{m/s}[/latex], respectively. If the object is moving at constant acceleration, what is the forcefulness interim on it?

A 120-kg astronaut is riding in a rocket sled that is sliding along an inclined plane. The sled has a horizontal component of acceleration of [latex]five.0\,\text{thou}\text{/}{\text{southward}}^{2}[/latex] and a downward component of [latex]3.8\,\text{m}\text{/}{\text{due south}}^{ii}[/latex]. Calculate the magnitude of the force on the rider by the sled. (Hint: Call back that gravitational acceleration must exist considered.)

Two forces are acting on a five.0-kg object that moves with acceleration [latex]2.0\,{\text{m/s}}^{2}[/latex] in the positive y-direction. If one of the forces acts in the positive 10-direction and has magnitude of 12 N, what is the magnitude of the other force?

Suppose that y'all are viewing a soccer game from a helicopter above the playing field. Ii soccer players simultaneously boot a stationary soccer brawl on the flat field; the soccer brawl has mass 0.420 kg. The kickoff player kicks with force 162 N at [latex]9.0^\circ[/latex] northward of west. At the same instant, the second actor kicks with force 215 N at [latex]15^\circ[/latex] due east of south. Find the acceleration of the ball in [latex]\mathbf{\hat{i}}[/latex] and [latex]\mathbf{\lid{j}}[/latex] form.

Evidence Solution

[latex]\mathbf{\overset{\to }{a}}=-248\mathbf{\hat{i}}-433\mathbf{\hat{j}}\text{thousand}\text{/}{\text{south}}^{two}[/latex]

A 10.0-kg mass hangs from a spring that has the spring abiding 535 N/thousand. Find the position of the terminate of the spring away from its residual position. (Utilise [latex]1000=nine.80\,{\text{m/south}}^{2}[/latex].)

A 0.0502-kg pair of fuzzy die is fastened to the rearview mirror of a car by a brusque string. The car accelerates at constant rate, and the dice hang at an angle of [latex]iii.xx^\circ[/latex] from the vertical considering of the car'due south acceleration. What is the magnitude of the acceleration of the car?

Evidence Solution

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

At a circus, a donkey pulls on a sled conveying a pocket-sized clown with a strength given by [latex]ii.48\mathbf{\hat{i}}+iv.33\mathbf{\lid{j}}\,\text{Northward}[/latex]. A equus caballus pulls on the same sled, aiding the hapless donkey, with a force of [latex]6.56\mathbf{\hat{i}}+5.33\mathbf{\lid{j}}\,\text{N}[/latex]. The mass of the sled is 575 kg. Using [latex]\mathbf{\chapeau{i}}[/latex] and [latex]\mathbf{\hat{j}}[/latex] form for the answer to each problem, find (a) the net strength on the sled when the 2 animals act together, (b) the dispatch of the sled, and (c) the velocity afterward vi.fifty s.

Hanging from the ceiling over a infant bed, well out of baby'due south achieve, is a string with plastic shapes, as shown hither. The string is taut (there is no slack), as shown past the straight segments. Each plastic shape has the same mass g, and they are equally spaced by a altitude d, as shown. The angles labeled [latex]\theta[/latex] draw the angle formed past the end of the string and the ceiling at each end. The centre length of sting is horizontal. The remaining two segments each grade an angle with the horizontal, labeled [latex]\varphi[/latex]. Let [latex]{T}_{one}[/latex] be the tension in the leftmost section of the string, [latex]{T}_{2}[/latex] be the tension in the department adjacent to information technology, and [latex]{T}_{three}[/latex] be the tension in the horizontal segment. (a) Notice 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 bending [latex]\theta[/latex]. (c) If [latex]\theta =v.ten^\circ[/latex], what is the value of [latex]\varphi[/latex]? (d) Find the distance 10 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}_{two}=\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]ii.56^\circ[/latex]; (d) [latex]10=d(2\,\text{cos}\,\theta +2\,\text{cos}(\text{arctan}(\frac{1}{ii}\text{tan}\,\theta ))+1)[/latex]

A bullet shot from a rifle has mass of 10.0 g and travels to the right at 350 grand/due south. It strikes a target, a big handbag of sand, penetrating it a distance of 34.0 cm. Find the magnitude and management of the retarding force that slows and stops the bullet.

An object is acted on by three simultaneous forces: [latex]{\mathbf{\overset{\to }{F}}}_{1}=(-iii.00\mathbf{\hat{i}}+2.00\mathbf{\hat{j}})\,\text{North}[/latex], [latex]{\mathbf{\overset{\to }{F}}}_{two}=(6.00\mathbf{\hat{i}}-4.00\mathbf{\lid{j}})\,\text{N}[/latex], and [latex]{\mathbf{\overset{\to }{F}}}_{3}=(2.00\mathbf{\hat{i}}+5.00\mathbf{\hat{j}})\,\text{Northward}[/latex]. The object experiences acceleration of [latex]4.23\,{\text{m/s}}^{two}[/latex]. (a) Discover the acceleration vector in terms of grand. (b) Find the mass of the object. (c) If the object begins from rest, find its speed afterward 5.00 s. (d) Detect the components of the velocity of the object after 5.00 s.

Show Solution

a. [latex]\mathbf{\overset{\to }{a}}=(\frac{5.00}{m}\mathbf{\hat{i}}+\frac{iii.00}{thou}\mathbf{\lid{j}})\,\text{m}\text{/}{\text{s}}^{ii};[/latex] b. one.38 kg; c. 21.ii m/southward; d. [latex]\mathbf{\overset{\to }{5}}=(18.i\mathbf{\chapeau{i}}+10.9\mathbf{\hat{j}})\,\text{yard}\text{/}{\text{s}}^{2}[/latex]

In a particle accelerator, a proton has mass [latex]ane.67\times {10}^{-27}\,\text{kg}[/latex] and an initial speed of [latex]2.00\times {x}^{5}\,\text{thou}\text{/}\text{s.}[/latex] It moves in a directly line, and its speed increases to [latex]9.00\times {10}^{5}\,\text{g}\text{/}\text{south}[/latex] in a distance of ten.0 cm. Assume that the acceleration is constant. Find the magnitude of the forcefulness exerted on the proton.

A drone is beingness directed across a frictionless ice-covered lake. The mass of the drone is 1.l kg, and its velocity is [latex]3.00\mathbf{\hat{i}}\text{m}\text{/}\text{s}[/latex]. After 10.0 due south, the velocity is [latex]nine.00\mathbf{\lid{i}}+4.00\mathbf{\hat{j}}\text{m}\text{/}\text{s}[/latex]. If a constant force in the horizontal direction is causing this change in motion, find (a) the components of the forcefulness and (b) the magnitude of the force.

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a. [latex]0.900\mathbf{\hat{i}}+0.600\mathbf{\chapeau{j}}\,\text{N}[/latex]; b. 1.08 N

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

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