How to Draw Free Body Diagrams in Statics

Learning Objectives

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

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

The first pace in describing and analyzing most phenomena in physics involves the careful drawing of a gratis-body diagram. Complimentary-body diagrams take been used in examples throughout this chapter. Remember that a free-trunk diagram must merely include the external forces acting on the trunk of involvement. Once nosotros have drawn an accurate free-trunk diagram, we can apply Newton'southward first law if the body is in equilibrium (balanced forces; that is, [latex] {F}_{\text{net}}=0 [/latex]) or Newton'south 2nd law if the trunk is accelerating (unbalanced force; that is, [latex] {F}_{\text{cyberspace}}\ne 0 [/latex]).

In Forces, we gave a cursory problem-solving strategy to help you understand free-body diagrams. Here, we add some details to the strategy that will help yous in constructing these diagrams.

Problem-Solving Strategy: Constructing Free-Body Diagrams

Observe the following rules when constructing a free-torso diagram:

1. Draw the object nether consideration; it does non take to be creative. At offset, you may want to describe a circumvolve effectually the object of involvement to be certain you focus on labeling the forces acting on the object. If you are treating the object equally a particle (no size or shape and no rotation), represent the object every bit a signal. We often identify this point at the origin of an xy-coordinate organisation.
2. Include all forces that act on the object, representing these forces as vectors. Consider the types of forces described in Mutual Forces—normal strength, friction, tension, and spring strength—besides as weight and practical strength. Do non include the net force on the object. With the exception of gravity, all of the forces we have discussed crave directly contact with the object. Notwithstanding, forces that the object exerts on its environment must not exist included. We never include both forces of an action-reaction pair.
3. Catechumen the free-body diagram into a more detailed diagram showing the x– and y-components of a given force (this is ofttimes helpful when solving a trouble using Newton's commencement or second constabulary). In this case, identify a squiggly line through the original vector to show that it is no longer in play—information technology has been replaced by its x– and y-components.
4. If at that place are two or more than objects, or bodies, in the problem, draw a separate free-trunk diagram for each object.

Annotation: If there is acceleration, we do not direct include it in the free-torso diagram; yet, it may assist to indicate dispatch outside the gratis-body diagram. You can label information technology in a different color to signal that it is dissever from the gratis-torso diagram.

Let's apply the problem-solving strategy in drawing a free-body diagram for a sled. In (Effigy)(a), a sled is pulled past strength P at an bending of [latex] xxx\text{°} [/latex]. In role (b), we show a free-body diagram for this situation, every bit described by steps ane and 2 of the problem-solving strategy. In function (c), we bear witness all forces in terms of their x– and y-components, in keeping with stride three.

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

Example

Two Blocks on an Inclined Plane

Construct the free-body diagram for object A and object B in (Figure).

Strategy

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

Solution

We start by creating a diagram for the first object of involvement. In (Figure)(a), object A is isolated (circled) and represented past a dot.

Figure five.32 (a) The free-trunk diagram for isolated object A. (b) The free-body diagram for isolated object B. Comparing the two drawings, we see that friction acts in the opposite management in the two figures. Because object A experiences a forcefulness that tends to pull information technology to the right, friction must act 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 act up the ramp. Friction always acts opposite the intended direction of motion.

We now include any force that acts on the body. Hither, no practical force is nowadays. The weight of the object acts every bit a force pointing vertically downward, and the presence of the cord indicates a force of tension pointing away from the object. Object A has 1 interface and hence experiences a normal force, 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 trend to slide downwardly, object A has a tendency to slide up with respect to the interface, so the friction [latex] {f}_{\text{BA}} [/latex] is directed downwardly parallel to the inclined plane.

As noted in stride 4 of the trouble-solving strategy, we then construct the complimentary-torso diagram in (Figure)(b) using the same approach. Object B experiences ii normal forces and two friction forces due to the presence of two contact surfaces. The interface with the inclined airplane exerts external forces of [latex] {N}_{\text{B}} [/latex] and [latex] {f}_{\text{B}} [/latex], and the interface with object B exerts the normal strength [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 role of this problem was circled in grayness. When yous are commencement learning how to depict free-body diagrams, you will notice it helpful to circle the object before deciding what forces are acting on that particular object. This focuses your attention, preventing you from considering forces that are non acting on the trunk.

Example

2 Blocks in Contact

A force is applied to ii blocks in contact, as shown.

Strategy

Describe a free-body diagram for each block. Be certain to consider Newton'southward tertiary police at the interface where the two blocks bear upon.

Solution

Significance[latex] {\overset{\to }{A}}_{21} [/latex] is the action force of block 2 on block 1. [latex] {\overset{\to }{A}}_{12} [/latex] is the reaction force of block i on cake ii. We use these gratuitous-body diagrams in Applications of Newton'due south Laws.

Case

Block on the Table (Coupled Blocks)

A cake rests on the table, every bit shown. A light rope is fastened to it and runs over a pulley. The other stop of the rope is attached to a second cake. The two blocks are said to be coupled. Block [latex] {m}_{2} [/latex] exerts a force due to its weight, which causes the organisation (2 blocks and a string) to accelerate.

Strategy

We presume that the string has no mass then that we practise not accept to consider it every bit a separate object. Draw a free-trunk diagram for each block.

Solution

Significance

Each cake accelerates (notice the labels shown for [latex] {\overset{\to }{a}}_{i} [/latex] and [latex] {\overset{\to }{a}}_{2} [/latex]); even so, assuming the string remains taut, they advance at the same rate. Thus, nosotros have [latex] {\overset{\to }{a}}_{ane}={\overset{\to }{a}}_{2} [/latex]. If we were to keep solving the problem, we could simply call the acceleration [latex] \overset{\to }{a} [/latex]. As well, we employ two complimentary-body diagrams because nosotros are usually finding tension T, which may require us to use a system of two equations in this type of trouble. The tension is the aforementioned on both [latex] {m}_{1}\,\text{and}\,{m}_{2} [/latex].

Check Your Understanding

(a) Draw the free-body diagram for the situation shown. (b) Redraw it showing components; use x-axes parallel to the two ramps.

Show Solution

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

View this simulation to predict, qualitatively, how an external force will touch on the speed and direction of an object's motion. Explain the effects with the help of a gratuitous-body diagram. Use free-torso diagrams to draw position, velocity, acceleration, and forcefulness graphs, and vice versa. Explain how the graphs relate to one another. Given a scenario or a graph, sketch all 4 graphs.

Summary

• To draw a gratuitous-body diagram, nosotros draw the object of interest, draw all forces acting on that object, and resolve all force vectors into x– and y-components. We must describe a separate complimentary-body diagram for each object in the trouble.
• A free-trunk diagram is a useful means of describing and analyzing all the forces that act on a body to decide equilibrium according to Newton's first police or dispatch according to Newton'southward second constabulary.

Fundamental Equations

Net external force	[latex] {\overset{\to }{F}}_{\text{net}}=\sum \overset{\to }{F}={\overset{\to }{F}}_{one}+{\overset{\to }{F}}_{2}+\text{⋯} [/latex]
Newton'south commencement police	[latex] \overset{\to }{five}=\,\text{constant when}\,{\overset{\to }{F}}_{\text{net}}=\overset{\to }{0}\,\text{N} [/latex]
Newton's second law, vector grade	[latex] {\overset{\to }{F}}_{\text{cyberspace}}=\sum \overset{\to }{F}=m\overset{\to }{a} [/latex]
Newton's second law, scalar form	[latex] {F}_{\text{net}}=ma [/latex]
Newton's 2nd law, component class	[latex] \sum {\overset{\to }{F}}_{ten}=m{\overset{\to }{a}}_{x}\text{,}\,\sum {\overset{\to }{F}}_{y}=m{\overset{\to }{a}}_{y},\,\text{and}\,\sum {\overset{\to }{F}}_{z}=k{\overset{\to }{a}}_{z}. [/latex]
Newton'due south second police, momentum course	[latex] {\overset{\to }{F}}_{\text{net}}=\frac{d\overset{\to }{p}}{dt} [/latex]
Definition of weight, vector course	[latex] \overset{\to }{w}=grand\overset{\to }{grand} [/latex]
Definition of weight, scalar course	[latex] w=mg [/latex]
Newton's third law	[latex] {\overset{\to }{F}}_{\text{AB}}=\text{−}{\overset{\to }{F}}_{\text{BA}} [/latex]
Normal force on an object resting on a horizontal surface, vector form	[latex] \overset{\to }{North}=\text{−}one thousand\overset{\to }{g} [/latex]
Normal force on an object resting on a horizontal surface, scalar form	[latex] N=mg [/latex]
Normal forcefulness on an object resting on an inclined aeroplane, scalar class	[latex] N=mg\text{cos}\,\theta [/latex]
Tension in a cablevision supporting an object of mass thousand at rest, scalar form	[latex] T=w=mg [/latex]

Conceptual Questions

In completing the solution for a problem involving forces, what do nosotros practise after amalgam the free-body diagram? That is, what practise nosotros apply?

If a book is located on a tabular array, how many forces should exist shown in a costless-body diagram of the book? Describe them.

Show Solution

Two forces of different types: weight interim downwards and normal force acting upwards

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

Problems

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

A car moves along a horizontal road. Draw a gratuitous-body diagram; be sure to include the friction of the road that opposes the forwards motion of the motorcar.

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 centre of his trunk, and include his weight.) (b) Requite a revised diagram showing the xy-component grade.

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

Show Solution

Boosted Problems

Ii small forces, [latex] {\overset{\to }{F}}_{one}=-2.40\hat{i}-6.10t\hat{j} [/latex] N and [latex] {\overset{\to }{F}}_{two}=8.50\hat{i}-nine.lxx\hat{j} [/latex] North, are exerted on a rogue asteroid by a pair of space tractors. (a) Notice the net force. (b) What are the magnitude and management of the net force? (c) If the mass of the asteroid is 125 kg, what dispatch does it experience (in vector form)? (d) What are the magnitude and direction of the dispatch?

Ii forces of 25 and 45 N act on an object. Their directions differ by [latex] 70\text{°} [/latex]. The resulting acceleration has magnitude of [latex] 10.0\,{\text{m/s}}^{2}. [/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\text{°} [/latex]. (a) What is the acceleration of the pianoforte up the ramp? (b) What is the velocity of the piano when it reaches the meridian if the ramp is 4.0 m long and the piano starts from rest?

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

Evidence Solution

For a swimmer who has simply jumped off a diving lath, assume air resistance is negligible. The swimmer has a mass of 80.0 kg and jumps off a board x.0 m in a higher place the water. 3 seconds after inbound the h2o, her downward motility is stopped. What average upward strength did the water exert on her?

(a) Find an equation to determine the magnitude of the internet force required to stop a automobile of mass thou, given that the initial speed of the automobile is [latex] {five}_{0} [/latex] and the stopping distance is x. (b) Notice the magnitude of the net force if the mass of the automobile is 1050 kg, the initial speed is 40.0 km/h, and the stopping altitude is 25.0 one thousand.

Show Solution

a. [latex] {F}_{\text{cyberspace}}=\frac{m({five}^{2}-{5}_{0}{}^{ii})}{2x} [/latex]; b. 2590 N

A sailboat has a mass of [latex] one.fifty\,×\,{10}^{three} [/latex] kg and is acted on by a force of [latex] ii.00\,×\,{x}^{iii} [/latex] Due north toward the east, while the wind acts backside the sails with a force of [latex] 3.00\,×\,{10}^{iii} [/latex] N in a direction [latex] 45\text{°} [/latex] northward of east. Find the magnitude and direction of the resulting acceleration.

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

Show Answer

[latex] \begin{assortment}{cc} {\overset{\to }{F}}_{\text{internet}}=4.05\chapeau{i}+12.0\hat{j}\text{N}\hfill \\ {\overset{\to }{F}}_{\text{internet}}=m\overset{\to }{a}⇒\overset{\to }{a}=0.405\hat{i}+1.20\hat{j}\,{\text{m/due south}}^{ii}\hfill \stop{array} [/latex]

A body of mass two.0 kg is moving along the 10-axis with a speed of iii.0 m/southward at the instant represented below. (a) What is the dispatch of the trunk? (b) What is the torso's velocity 10.0 s later? (c) What is its displacement after 10.0 southward?

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

Shown below is a body of mass 1.0 kg under the influence of the forces [latex] {\overset{\to }{F}}_{A} [/latex], [latex] {\overset{\to }{F}}_{B} [/latex], and [latex] m\overset{\to }{grand} [/latex]. If the body accelerates to the left at [latex] 20\,{\text{m/s}}^{2} [/latex], what are [latex] {\overset{\to }{F}}_{A} [/latex] and [latex] {\overset{\to }{F}}_{B} [/latex]?

A force acts on a machine of mass m so that the speed five of the motorcar increases with position x equally [latex] v=k{x}^{2} [/latex], where one thousand is constant and all quantities are in SI units. Find the strength interim on the motorcar as a function of position.

Evidence Solution

[latex] F=2kmx [/latex]; First, take the derivative of the velocity part to obtain [latex] a=2kx [/latex]. Then apply Newton's 2d law [latex] F=ma=1000(2kx)=2kmx [/latex].

A seven.0-N force parallel to an incline is applied to a i.0-kg crate. The ramp is tilted at [latex] 20\text{°} [/latex] and is frictionless. (a) What is the acceleration of the crate? (b) If all other conditions are the aforementioned only the ramp has a friction force of 1.9 N, what is the acceleration?

2 boxes, A and B, are at remainder. Box A is on level ground, while box B rests on an inclined airplane tilted at angle [latex] \theta [/latex] with the horizontal. (a) Write expressions for the normal force interim on each block. (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\text{°} [/latex], which force is greater?

Evidence Solution

a. For box A, [latex] {Northward}_{\text{A}}=mg [/latex] and [latex] {N}_{\text{B}}=mg\,\text{cos}\,\theta [/latex]; b. [latex] {N}_{\text{A}}>{Northward}_{\text{B}} [/latex] because for [latex] \theta <90\text{°} [/latex], [latex] \text{cos}\,\theta <1 [/latex]; c. [latex] {N}_{\text{A}}>{N}_{\text{B}} [/latex] when [latex] \theta =10\text{°} [/latex]

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

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

Evidence Solution

a. eight.66 N; b. 0.433 thousand

Shown below is a 30.0-kg cake resting on a frictionless ramp inclined at [latex] 60\text{°} [/latex] to the horizontal. The block is held by a spring that is stretched 5.0 cm. What is the forcefulness constant of the spring?

In building a firm, carpenters utilize nails from a large box. The box is suspended from a leap twice during the twenty-four hours to mensurate the usage of nails. At the beginning of the day, the spring stretches fifty cm. At the end of the day, the spring stretches thirty cm. What fraction or percentage of the nails have been used?

Show Solution

0.40 or 40%

A forcefulness is applied to a block to move it up a [latex] 30\text{°} [/latex] incline. The incline is frictionless. If [latex] F=65.0\,\text{Northward} [/latex] and [latex] Grand=5.00\,\text{kg} [/latex], what is the magnitude of the acceleration of the block?

Two forces are practical to a 5.0-kg object, and it accelerates at a charge per unit of [latex] 2.0\,{\text{m/s}}^{two} [/latex] in the positive y-direction. If one of the forces acts in the positive x-management with magnitude 12.0 North, notice the magnitude of the other strength.

The block on the right shown below has more mass than the block on the left ([latex] {m}_{2}>{m}_{1} [/latex]). Draw complimentary-torso diagrams for each block.

Challenge Issues

If two tugboats pull on a disabled vessel, as shown here in an overhead view, the disabled vessel will be pulled along the direction indicated by the issue of the exerted forces. (a) Draw a free-body diagram for the vessel. Presume no friction or drag forces affect the vessel. (b) Did you include all forces in the overhead view in your free-body diagram? Why or why not?

A 10.0-kg object is initially moving eastward at xv.0 one thousand/southward. So a force acts on it for 2.00 s, after which information technology moves northwest, likewise at 15.0 m/s. What are the magnitude and direction of the boilerplate forcefulness that acted on the object over the 2.00-southward interval?

On June 25, 1983, shot-putter Udo Beyer of East Germany threw the seven.26-kg shot 22.22 m, which at that time was a world record. (a) If the shot was released at a height of two.20 m with a projection angle of [latex] 45.0\text{°} [/latex], what was its initial velocity? (b) If while in Beyer'south manus the shot was accelerated uniformly over a distance of 1.20 m, what was the cyberspace force on it?

Show Solution

a. fourteen.ane m/s; b. 601 N

A trunk of mass m moves in a horizontal direction such that at time t its position is given by [latex] ten(t)=a{t}^{4}+b{t}^{3}+ct, [/latex] where a, b, and c are constants. (a) What is the acceleration of the body? (b) What is the time-dependent forcefulness interim on the body?

A body of mass m has initial velocity [latex] {v}_{0} [/latex] in the positive x-management. It is acted on by a constant force F for time t until the velocity becomes zero; the force continues to act on the body until its velocity becomes [latex] \text{−}{v}_{0} [/latex] in the same corporeality of fourth dimension. Write an expression for the total 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=6.0\,\text{s} [/latex] and [latex] t=8.0\,\text{s} [/latex] are [latex] (three.0\lid{i}-half-dozen.0\hat{j}+iv.0\hat{thousand})\,\text{one thousand/s} [/latex] and [latex] (-2.0\hat{i}+4.0\chapeau{grand})\,\text{yard/southward} [/latex], respectively. If the object is moving at constant acceleration, what is the forcefulness interim on information technology?

A 120-kg astronaut is riding in a rocket sled that is sliding along an inclined aeroplane. The sled has a horizontal component of dispatch of [latex] 5.0\,\text{g}\text{/}{\text{due south}}^{2} [/latex] and a downward component of [latex] 3.8\,\text{g}\text{/}{\text{s}}^{2} [/latex]. Calculate the magnitude of the forcefulness on the rider by the sled. (Hint: Remember that gravitational dispatch must be considered.)

Two forces are interim on a five.0-kg object that moves with acceleration [latex] ii.0\,{\text{g/southward}}^{ii} [/latex] in the positive y-direction. If i of the forces acts in the positive ten-direction and has magnitude of 12 North, what is the magnitude of the other force?

Suppose that you are viewing a soccer game from a helicopter above the playing field. 2 soccer players simultaneously kick a stationary soccer ball on the flat field; the soccer brawl has mass 0.420 kg. The outset actor kicks with force 162 N at [latex] 9.0\text{°} [/latex] north of west. At the aforementioned instant, the second player kicks with force 215 N at [latex] 15\text{°} [/latex] east of s. Find the acceleration of the ball in [latex] \chapeau{i} [/latex] and [latex] \chapeau{j} [/latex] form.

Show Solution

[latex] [/latex][latex] \overset{\to }{a}=-248\hat{i}-433\hat{j}\text{m}\text{/}{\text{southward}}^{2} [/latex]

A 10.0-kg mass hangs from a spring that has the bound abiding 535 N/1000. Detect the position of the end of the spring away from its rest position. (Use [latex] g=9.80\,{\text{m/s}}^{2} [/latex].)

A 0.0502-kg pair of fuzzy dice is fastened to the rearview mirror of a machine by a short string. The car accelerates at constant rate, and the dice hang at an angle of [latex] 3.20\text{°} [/latex] from the vertical because of the car's acceleration. What is the magnitude of the acceleration of the automobile?

Evidence Solution

[latex] 0.548\,{\text{m/s}}^{2} [/latex]

At a circus, a donkey pulls on a sled conveying a small clown with a force given by [latex] 2.48\hat{i}+4.33\hat{j}\,\text{N} [/latex]. A horse pulls on the same sled, aiding the hapless donkey, with a strength of [latex] six.56\hat{i}+5.33\hat{j}\,\text{Due north} [/latex]. The mass of the sled is 575 kg. Using [latex] \chapeau{i} [/latex] and [latex] \hat{j} [/latex] course for the answer to each problem, find (a) the net force on the sled when the two animals act together, (b) the acceleration of the sled, and (c) the velocity after 6.50 s.

Hanging from the ceiling over a baby bed, well out of babe'due south reach, is a string with plastic shapes, every bit shown here. The string is taut (at that place is no slack), every bit shown past the direct segments. Each plastic shape has the same mass 1000, and they are equally spaced by a altitude d, as shown. The angles labeled [latex] \theta [/latex] depict the angle formed past the terminate of the string and the ceiling at each cease. The center length of sting is horizontal. The remaining two segments each form an angle with the horizontal, labeled [latex] \varphi [/latex]. Let [latex] {T}_{i} [/latex] exist the tension in the leftmost section of the string, [latex] {T}_{2} [/latex] be the tension in the department adjacent to it, and [latex] {T}_{3} [/latex] exist the tension in the horizontal segment. (a) Detect an equation for the tension in each section of the cord in terms of the variables m, g, and [latex] \theta [/latex]. (b) Find the bending [latex] \varphi [/latex] in terms of the angle [latex] \theta [/latex]. (c) If [latex] \theta =5.10\text{°} [/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}_{ane}=\frac{2mg}{\text{sin}\,\theta } [/latex], [latex] {T}_{ii}=\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{one}{ii}\text{tan}\,\theta ) [/latex]; c. [latex] 2.56\text{°} [/latex]; (d) [latex] ten=d(ii\,\text{cos}\,\theta +2\,\text{cos}(\text{arctan}(\frac{i}{2}\text{tan}\,\theta ))+ane) [/latex]

A bullet shot from a rifle has mass of 10.0 thou and travels to the right at 350 one thousand/s. It strikes a target, a large handbag of sand, penetrating it a distance of 34.0 cm. Find the magnitude and direction of the retarding force that slows and stops the bullet.

An object is acted on by three simultaneous forces: [latex] {\overset{\to }{F}}_{one}=(-iii.00\chapeau{i}+two.00\hat{j})\,\text{Due north} [/latex], [latex] {\overset{\to }{F}}_{ii}=(vi.00\chapeau{i}-4.00\hat{j})\,\text{N} [/latex], and [latex] {\overset{\to }{F}}_{3}=(two.00\chapeau{i}+five.00\hat{j})\,\text{Northward} [/latex]. The object experiences acceleration of [latex] 4.23\,{\text{m/south}}^{2} [/latex]. (a) Discover the acceleration vector in terms of thousand. (b) Find the mass of the object. (c) If the object begins from residue, notice its speed after 5.00 s. (d) Find the components of the velocity of the object subsequently 5.00 s.

Show Solution

a. [latex] \overset{\to }{a}=(\frac{5.00}{m}\hat{i}+\frac{3.00}{thousand}\chapeau{j})\,\text{yard}\text{/}{\text{s}}^{2}; [/latex] b. i.38 kg; c. 21.2 grand/southward; d. [latex] \overset{\to }{v}=(18.1\hat{i}+10.9\chapeau{j})\,\text{m}\text{/}{\text{s}}^{two} [/latex]

In a particle accelerator, a proton has mass [latex] 1.67\,×\,{x}^{-27}\,\text{kg} [/latex] and an initial speed of [latex] 2.00\,×\,{10}^{5}\,\text{m}\text{/}\text{s.} [/latex] Information technology moves in a straight line, and its speed increases to [latex] nine.00\,×\,{ten}^{5}\,\text{one thousand}\text{/}\text{s} [/latex] in a distance of 10.0 cm. Assume that the acceleration is abiding. Find 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 ane.50 kg, and its velocity is [latex] 3.00\lid{i}\text{thousand}\text{/}\text{s} [/latex]. Afterward ten.0 southward, the velocity is [latex] 9.00\hat{i}+4.00\hat{j}\text{m}\text{/}\text{s} [/latex]. If a abiding force in the horizontal direction is causing this modify in motion, detect (a) the components of the force and (b) the magnitude of the strength.

Show Solution

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

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Source: https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/5-7-drawing-free-body-diagrams/

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