Draw a Free Body Diagram of the Car
Learning Objectives
By the end of the section, you will be able to:
- Explicate the rules for drawing a free-body diagram
- Construct gratuitous-body diagrams for different situations
The first stride in describing and analyzing most phenomena in physics involves the careful cartoon of a free-torso diagram. Gratis-trunk diagrams have been used in examples throughout this chapter. Remember that a free-body diagram must only include the external forces acting on the trunk of interest. One time we have drawn an accurate free-body diagram, nosotros tin can apply Newton'south starting time constabulary if the torso is in equilibrium (balanced forces; that is, [latex] {F}_{\text{net}}=0 [/latex]) or Newton's second constabulary if the body is accelerating (unbalanced forcefulness; that is, [latex] {F}_{\text{net}}\ne 0 [/latex]).
In Forces, we gave a brief trouble-solving strategy to help you understand complimentary-trunk diagrams. Here, we add some details to the strategy that volition assist yous in amalgam these diagrams.
Problem-Solving Strategy: Constructing Complimentary-Torso Diagrams
Observe the following rules when constructing a gratis-trunk diagram:
- Draw the object under consideration; information technology does not take to be creative. At offset, you may want to describe a circle effectually the object of interest to be sure you focus on labeling the forces acting on the object. If y'all are treating the object as a particle (no size or shape and no rotation), represent the object equally a point. Nosotros often place this signal at the origin of an xy-coordinate system.
- Include all forces that human activity on the object, representing these forces every bit vectors. Consider the types of forces described in Common Forces—normal force, friction, tension, and leap force—as well as weight and applied force. Do not include the net forcefulness on the object. With the exception of gravity, all of the forces nosotros accept discussed require direct contact with the object. However, forces that the object exerts on its surround must not be included. Nosotros never include both forces of an action-reaction pair.
- 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 commencement or second law). In this instance, 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.
- If there are two or more objects, or bodies, in the problem, draw a divide free-body diagram for each object.
Note: If at that place is acceleration, we exercise non directly include information technology in the gratis-torso diagram; still, information technology may assistance to indicate acceleration exterior the free-body diagram. Y'all can label information technology in a different colour to signal that it is separate from the gratis-body diagram.
Allow'south apply the problem-solving strategy in drawing a gratis-body diagram for a sled. In (Effigy)(a), a sled is pulled past force P at an angle of [latex] xxx\text{°} [/latex]. In part (b), we bear witness a free-body diagram for this situation, every bit described by steps 1 and 2 of the trouble-solving strategy. In part (c), we show all forces in terms of their x– and y-components, in keeping with stride 3.
Effigy five.31 (a) A moving sled is shown as (b) a gratuitous-body diagram and (c) a complimentary-body diagram with force components.
Example
Ii 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
Nosotros kickoff by creating a diagram for the first object of involvement. In (Figure)(a), object A is isolated (circled) and represented past a dot.
Effigy five.32 (a) The costless-torso diagram for isolated object A. (b) The complimentary-torso diagram for isolated object B. Comparison the two drawings, we run into that friction acts in the opposite direction in the two figures. Because object A experiences a force that tends to pull it to the right, friction must act to the left. Considering object B experiences a component of its weight that pulls information technology to the left, down the incline, the friction force must oppose information technology and act up the ramp. Friction always acts opposite the intended direction of motion.
We now include whatsoever force that acts on the body. Hither, no applied force is present. The weight of the object acts as a forcefulness pointing vertically downward, and the presence of the cord indicates a force of tension pointing away from the object. Object A has one interface and hence experiences a normal force, directed away from the interface. The source of this force is object B, and this normal force is labeled accordingly. Since object B has a trend to slide down, object A has a trend to slide upwards with respect to the interface, so the friction [latex] {f}_{\text{BA}} [/latex] is directed downward parallel to the inclined airplane.
As noted in footstep iv of the problem-solving strategy, nosotros then construct the free-body diagram in (Figure)(b) using the same approach. Object B experiences two normal forces and ii friction forces due to the presence of 2 contact surfaces. The interface with the inclined plane exerts external forces of [latex] {North}_{\text{B}} [/latex] and [latex] {f}_{\text{B}} [/latex], and the interface with object B exerts the normal force [latex] {North}_{\text{AB}} [/latex] and friction [latex] {f}_{\text{AB}} [/latex]; [latex] {N}_{\text{AB}} [/latex] is directed abroad from object B, and [latex] {f}_{\text{AB}} [/latex] is opposing the tendency of the relative move of object B with respect to object A.
Significance
The object nether consideration in each part of this trouble was circled in greyness. When you lot are first learning how to draw gratis-body diagrams, you volition find it helpful to circumvolve the object before deciding what forces are interim on that particular object. This focuses your attending, preventing y'all from considering forces that are not interim on the body.
Example
Two Blocks in Contact
A force is applied to two blocks in contact, as shown.
Strategy
Draw a free-body diagram for each block. Exist sure to consider Newton'south tertiary police at the interface where the two blocks touch.
Solution
Significance[latex] {\overset{\to }{A}}_{21} [/latex] is the activeness force of block two on block 1. [latex] {\overset{\to }{A}}_{12} [/latex] is the reaction forcefulness of block ane on block 2. We use these free-torso diagrams in Applications of Newton's Laws.
Example
Block on the Table (Coupled Blocks)
A block rests on the tabular array, equally shown. A light rope is attached 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 system (ii blocks and a string) to accelerate.
Strategy
Nosotros assume that the cord has no mass so that nosotros do non have to consider it as a separate object. Draw a costless-body diagram for each cake.
Solution
Significance
Each block accelerates (notice the labels shown for [latex] {\overset{\to }{a}}_{1} [/latex] and [latex] {\overset{\to }{a}}_{2} [/latex]); however, assuming the string remains taut, they accelerate at the same rate. Thus, nosotros have [latex] {\overset{\to }{a}}_{1}={\overset{\to }{a}}_{ii} [/latex]. If we were to continue solving the trouble, we could simply call the dispatch [latex] \overset{\to }{a} [/latex]. Likewise, we use two free-body diagrams because we are normally finding tension T, which may require us to use a organization of two equations in this type of problem. The tension is the same on both [latex] {thousand}_{ane}\,\text{and}\,{1000}_{ii} [/latex].
Check Your Agreement
(a) Depict the complimentary-torso diagram for the state of affairs shown. (b) Redraw information technology showing components; utilize x-axes parallel to the two ramps.
Bear witness Solution
Figure a shows a costless body diagram of an object on a line that slopes down to the right. Arrow T from the object points right and upward, parallel to the slope. Arrow N1 points left and upward, perpendicular to the gradient. Arrow w1 points vertically down. Arrow w1x points left and down, parallel to the slope. Arrow w1y points right and down, perpendicular to the slope. Figure b shows a free body diagram of an object on a line that slopes downwards to the left. Arrow N2 from the object points right and up, perpendicular to the gradient. Arrow T points left and upward, parallel to the slope. Arrow w2 points vertically down. Arrow w2y points left and down, perpendicular to the slope. Pointer w2x points correct and down, parallel to the slope.
View this simulation to predict, qualitatively, how an external strength volition affect the speed and management of an object'south motion. Explicate the effects with the help of a free-body diagram. Use free-body diagrams to draw position, velocity, acceleration, and force graphs, and vice versa. Explicate how the graphs relate to one another. Given a scenario or a graph, sketch all four graphs.
Summary
- To draw a free-body diagram, we draw the object of involvement, draw all forces acting on that object, and resolve all force vectors into x– and y-components. We must draw a separate free-body diagram for each object in the trouble.
- A costless-trunk 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'south first police force or acceleration according to Newton's 2nd law.
Central Equations
| Internet external force | [latex] {\overset{\to }{F}}_{\text{net}}=\sum \overset{\to }{F}={\overset{\to }{F}}_{one}+{\overset{\to }{F}}_{2}+\text{⋯} [/latex] |
| Newton's get-go law | [latex] \overset{\to }{v}=\,\text{constant when}\,{\overset{\to }{F}}_{\text{net}}=\overset{\to }{0}\,\text{Northward} [/latex] |
| Newton's second police, vector grade | [latex] {\overset{\to }{F}}_{\text{net}}=\sum \overset{\to }{F}=g\overset{\to }{a} [/latex] |
| Newton's second police, scalar form | [latex] {F}_{\text{net}}=ma [/latex] |
| Newton's 2nd law, component form | [latex] \sum {\overset{\to }{F}}_{x}=m{\overset{\to }{a}}_{x}\text{,}\,\sum {\overset{\to }{F}}_{y}=k{\overset{\to }{a}}_{y},\,\text{and}\,\sum {\overset{\to }{F}}_{z}=g{\overset{\to }{a}}_{z}. [/latex] |
| Newton'due south 2nd law, momentum form | [latex] {\overset{\to }{F}}_{\text{net}}=\frac{d\overset{\to }{p}}{dt} [/latex] |
| Definition of weight, vector form | [latex] \overset{\to }{w}=m\overset{\to }{thou} [/latex] |
| Definition of weight, scalar form | [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 }{Due north}=\text{−}m\overset{\to }{g} [/latex] |
| Normal force on an object resting on a horizontal surface, scalar form | [latex] N=mg [/latex] |
| Normal force on an object resting on an inclined aeroplane, scalar form | [latex] N=mg\text{cos}\,\theta [/latex] |
| Tension in a cable supporting an object of mass thousand at residue, scalar form | [latex] T=w=mg [/latex] |
Conceptual Questions
In completing the solution for a trouble involving forces, what do we do after constructing the free-body diagram? That is, what do nosotros apply?
If a book is located on a table, how many forces should be shown in a free-body diagram of the book? Describe them.
Show Solution
Ii forces of unlike types: weight acting downwardly and normal force interim upwardly
If the book in the previous question is in complimentary fall, how many forces should be shown in a free-trunk diagram of the volume? Describe them.
Problems
A ball of mass m hangs at rest, suspended past a cord. (a) Sketch all forces. (b) Describe the free-torso diagram for the ball.
A automobile moves along a horizontal road. Describe a free-trunk diagram; exist sure to include the friction of the road that opposes the forward motility of the car.
Show Solution
A runner pushes against the runway, as shown. (a) Provide a gratis-body diagram showing all the forces on the runner. (Hint: Place all forces at the center of his body, and include his weight.) (b) Give a revised diagram showing the xy-component form.
The traffic light hangs from the cables equally shown. Draw a free-torso diagram on a coordinate plane for this situation.
Prove Solution
Additional Problems
Ii small-scale forces, [latex] {\overset{\to }{F}}_{1}=-two.40\lid{i}-vi.10t\hat{j} [/latex] N and [latex] {\overset{\to }{F}}_{2}=8.fifty\hat{i}-ix.lxx\lid{j} [/latex] N, are exerted on a rogue asteroid by a pair of infinite tractors. (a) Observe 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 feel (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] seventy\text{°} [/latex]. The resulting acceleration has magnitude of [latex] x.0\,{\text{thousand/south}}^{2}. [/latex] What is the mass of the body?
A force of 1600 N acts parallel to a ramp to push a 300-kg piano into a moving van. The ramp is inclined at [latex] twenty\text{°} [/latex]. (a) What is the acceleration of the pianoforte upwardly the ramp? (b) What is the velocity of the pianoforte when it reaches the summit if the ramp is 4.0 m long and the piano starts from residual?
Draw a free-torso diagram of a diver who has entered the water, moved downward, and is acted on past an upwards forcefulness due to the water which balances the weight (that is, the diver is suspended).
Bear witness Solution 
For a swimmer who has just jumped off a diving board, assume air resistance is negligible. The swimmer has a mass of 80.0 kg and jumps off a lath ten.0 m above the h2o. Three seconds subsequently inbound the water, her downward motion is stopped. What average upwards force did the water exert on her?
(a) Find an equation to determine the magnitude of the internet forcefulness required to terminate a car of mass m, given that the initial speed of the car is [latex] {v}_{0} [/latex] and the stopping altitude is 10. (b) Find the magnitude of the net force if the mass of the machine is 1050 kg, the initial speed is xl.0 km/h, and the stopping distance is 25.0 g.
Show Solution
a. [latex] {F}_{\text{net}}=\frac{m({v}^{ii}-{v}_{0}{}^{2})}{2x} [/latex]; b. 2590 N
A sailboat has a mass of [latex] one.l\,×\,{x}^{3} [/latex] kg and is acted on by a strength of [latex] ii.00\,×\,{10}^{3} [/latex] N toward the east, while the current of air acts backside the sails with a strength of [latex] 3.00\,×\,{10}^{3} [/latex] North in a direction [latex] 45\text{°} [/latex] north of east. Find the magnitude and direction of the resulting acceleration.
Notice the acceleration of the body of mass 10.0 kg shown beneath.
Evidence Respond
[latex] \begin{assortment}{cc} {\overset{\to }{F}}_{\text{cyberspace}}=4.05\lid{i}+12.0\hat{j}\text{N}\hfill \\ {\overset{\to }{F}}_{\text{net}}=m\overset{\to }{a}⇒\overset{\to }{a}=0.405\lid{i}+one.20\chapeau{j}\,{\text{thou/due south}}^{ii}\hfill \end{array} [/latex]
A torso of mass 2.0 kg is moving along the x-axis with a speed of 3.0 yard/s at the instant represented beneath. (a) What is the acceleration of the body? (b) What is the body'due south velocity 10.0 s later on? (c) What is its deportation after 10.0 s?
Force [latex] {\overset{\to }{F}}_{\text{B}} [/latex] has twice the magnitude of forcefulness [latex] {\overset{\to }{F}}_{\text{A}}. [/latex] Find the direction in which the particle accelerates in this figure.
Shown beneath 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 }{g} [/latex]. If the body accelerates to the left at [latex] 20\,{\text{1000/s}}^{2} [/latex], what are [latex] {\overset{\to }{F}}_{A} [/latex] and [latex] {\overset{\to }{F}}_{B} [/latex]?
A forcefulness acts on a automobile of mass g and so that the speed v of the auto increases with position 10 as [latex] 5=yard{x}^{2} [/latex], where thousand is constant and all quantities are in SI units. Find the force acting on the car every bit a part of position.
Show Solution
[latex] F=2kmx [/latex]; Outset, accept the derivative of the velocity function to obtain [latex] a=2kx [/latex]. Then apply Newton's second law [latex] F=ma=m(2kx)=2kmx [/latex].
A seven.0-N strength parallel to an incline is applied to a 1.0-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 weather are the aforementioned simply the ramp has a friction force of 1.ix N, what is the acceleration?
2 boxes, A and B, are at residual. Box A is on level footing, while box B rests on an inclined plane tilted at angle [latex] \theta [/latex] with the horizontal. (a) Write expressions for the normal forcefulness acting 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 bending of incline is [latex] 10\text{°} [/latex], which force is greater?
Show Solution
a. For box A, [latex] {N}_{\text{A}}=mg [/latex] and [latex] {N}_{\text{B}}=mg\,\text{cos}\,\theta [/latex]; b. [latex] {N}_{\text{A}}>{North}_{\text{B}} [/latex] because for [latex] \theta <xc\text{°} [/latex], [latex] \text{cos}\,\theta <one [/latex]; c. [latex] {N}_{\text{A}}>{N}_{\text{B}} [/latex] when [latex] \theta =10\text{°} [/latex]
A mass of 250.0 g is suspended from a bound hanging vertically. The spring stretches 6.00 cm. How much will the spring stretch if the suspended mass is 530.0 g?
As shown below, two identical springs, each with the spring abiding 20 N/yard, support a fifteen.0-N weight. (a) What is the tension in bound A? (b) What is the amount of stretch of spring A from the rest position?
Show Solution
a. 8.66 N; b. 0.433 1000
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 force abiding of the spring?
In edifice a house, carpenters employ nails from a large box. The box is suspended from a spring twice during the day to measure the usage of nails. At the beginning of the day, the spring stretches 50 cm. At the end of the day, the spring stretches thirty cm. What fraction or per centum of the nails take been used?
Testify Solution
0.forty or twoscore%
A force is applied to a block to movement it up a [latex] 30\text{°} [/latex] incline. The incline is frictionless. If [latex] F=65.0\,\text{N} [/latex] and [latex] 1000=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 information technology accelerates at a charge per unit of [latex] ii.0\,{\text{m/s}}^{2} [/latex] in the positive y-direction. If ane of the forces acts in the positive ten-direction with magnitude 12.0 N, discover 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}>{m}_{ane} [/latex]). Draw gratuitous-body diagrams for each block.
Challenge Problems
If 2 tugboats pull on a disabled vessel, as shown hither in an overhead view, the disabled vessel will exist pulled along the management indicated by the result of the exerted forces. (a) Depict a costless-torso diagram for the vessel. Assume no friction or elevate forces touch on the vessel. (b) Did you include all forces in the overhead view in your gratis-body diagram? Why or why not?
A ten.0-kg object is initially moving east at 15.0 m/s. Then a strength acts on it for 2.00 southward, after which it moves northwest, likewise at 15.0 thou/s. What are the magnitude and management of the average 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 7.26-kg shot 22.22 m, which at that fourth dimension was a world tape. (a) If the shot was released at a peak of 2.20 m with a project angle of [latex] 45.0\text{°} [/latex], what was its initial velocity? (b) If while in Beyer'due south mitt the shot was accelerated uniformly over a distance of 1.20 m, what was the net forcefulness on it?
Show Solution
a. 14.1 m/s; b. 601 Northward
A torso of mass yard moves in a horizontal direction such that at fourth dimension t its position is given past [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 time-dependent force acting on the body?
A body of mass m has initial velocity [latex] {five}_{0} [/latex] in the positive x-management. Information technology is acted on by a constant force F for time t until the velocity becomes zero; the force continues to human activity on the body until its velocity becomes [latex] \text{−}{v}_{0} [/latex] in the aforementioned amount of time. Write an expression for the total distance the trunk travels in terms of the variables indicated.
Show Solution
[latex] \frac{F}{m}{t}^{2} [/latex]
The velocities of a 3.0-kg object at [latex] t=6.0\,\text{south} [/latex] and [latex] t=eight.0\,\text{s} [/latex] are [latex] (three.0\lid{i}-6.0\hat{j}+4.0\hat{one thousand})\,\text{grand/s} [/latex] and [latex] (-ii.0\hat{i}+4.0\hat{m})\,\text{m/s} [/latex], respectively. If the object is moving at constant dispatch, what is the force acting on it?
A 120-kg astronaut is riding in a rocket sled that is sliding forth an inclined airplane. The sled has a horizontal component of acceleration of [latex] v.0\,\text{m}\text{/}{\text{s}}^{2} [/latex] and a down component of [latex] three.eight\,\text{m}\text{/}{\text{s}}^{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 interim on a five.0-kg object that moves with dispatch [latex] two.0\,{\text{k/southward}}^{two} [/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 y'all are viewing a soccer game from a helicopter above the playing field. 2 soccer players simultaneously kicking a stationary soccer ball on the flat field; the soccer brawl has mass 0.420 kg. The commencement player kicks with force 162 Northward at [latex] 9.0\text{°} [/latex] north of west. At the same instant, the second player kicks with strength 215 N at [latex] 15\text{°} [/latex] due east of south. Find the acceleration of the ball in [latex] \chapeau{i} [/latex] and [latex] \hat{j} [/latex] form.
Show Solution
[latex] [/latex][latex] \overset{\to }{a}=-248\hat{i}-433\hat{j}\text{m}\text{/}{\text{s}}^{2} [/latex]
A 10.0-kg mass hangs from a leap that has the spring constant 535 N/grand. Observe the position of the end of the leap away from its rest position. (Apply [latex] chiliad=ix.80\,{\text{g/south}}^{2} [/latex].)
A 0.0502-kg pair of fuzzy dice is fastened to the rearview mirror of a car by a short cord. The car accelerates at constant rate, and the dice hang at an angle of [latex] three.20\text{°} [/latex] from the vertical because of the car'southward acceleration. What is the magnitude of the acceleration of the car?
Show Solution
[latex] 0.548\,{\text{chiliad/s}}^{2} [/latex]
At a circus, a ass pulls on a sled carrying a small clown with a force given by [latex] ii.48\hat{i}+iv.33\chapeau{j}\,\text{North} [/latex]. A equus caballus pulls on the aforementioned sled, aiding the hapless donkey, with a force of [latex] half-dozen.56\lid{i}+five.33\chapeau{j}\,\text{North} [/latex]. The mass of the sled is 575 kg. Using [latex] \lid{i} [/latex] and [latex] \hat{j} [/latex] form for the respond to each trouble, notice (a) the net strength 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 infant's reach, is a string with plastic shapes, as shown here. The string is taut (at that place is no slack), equally shown past the straight segments. Each plastic shape has the same mass m, and they are as spaced by a distance d, as shown. The angles labeled [latex] \theta [/latex] describe the bending formed past the end of the string and the ceiling at each end. The center length of sting is horizontal. The remaining two segments each class an bending with the horizontal, labeled [latex] \varphi [/latex]. Permit [latex] {T}_{1} [/latex] exist the tension in the leftmost section of the string, [latex] {T}_{ii} [/latex] be the tension in the section side by side 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) Notice the bending [latex] \varphi [/latex] in terms of the bending [latex] \theta [/latex]. (c) If [latex] \theta =v.10\text{°} [/latex], what is the value of [latex] \varphi [/latex]? (d) Notice the distance x betwixt the endpoints in terms of d and [latex] \theta [/latex].
Show Solution
a. [latex] {T}_{one}=\frac{2mg}{\text{sin}\,\theta } [/latex], [latex] {T}_{2}=\frac{mg}{\text{sin}(\text{arctan}(\frac{1}{two}\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\text{°} [/latex]; (d) [latex] x=d(2\,\text{cos}\,\theta +two\,\text{cos}(\text{arctan}(\frac{1}{2}\text{tan}\,\theta ))+1) [/latex]
A bullet shot from a burglarize has mass of 10.0 k and travels to the correct at 350 grand/s. It strikes a target, a large bag 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 past 3 simultaneous forces: [latex] {\overset{\to }{F}}_{1}=(-3.00\hat{i}+2.00\hat{j})\,\text{Due north} [/latex], [latex] {\overset{\to }{F}}_{2}=(6.00\chapeau{i}-4.00\chapeau{j})\,\text{Due north} [/latex], and [latex] {\overset{\to }{F}}_{three}=(2.00\hat{i}+five.00\lid{j})\,\text{Due north} [/latex]. The object experiences dispatch of [latex] iv.23\,{\text{thou/due south}}^{2} [/latex]. (a) Find the acceleration vector in terms of m. (b) Find the mass of the object. (c) If the object begins from rest, find its speed after 5.00 s. (d) Detect the components of the velocity of the object after 5.00 south.
Evidence Solution
a. [latex] \overset{\to }{a}=(\frac{5.00}{grand}\hat{i}+\frac{3.00}{thou}\hat{j})\,\text{m}\text{/}{\text{s}}^{2}; [/latex] b. ane.38 kg; c. 21.2 m/s; d. [latex] \overset{\to }{v}=(18.i\lid{i}+x.ix\hat{j})\,\text{g}\text{/}{\text{s}}^{two} [/latex]
In a particle accelerator, a proton has mass [latex] 1.67\,×\,{10}^{-27}\,\text{kg} [/latex] and an initial speed of [latex] two.00\,×\,{10}^{5}\,\text{chiliad}\text{/}\text{s.} [/latex] It moves in a straight line, and its speed increases to [latex] 9.00\,×\,{10}^{5}\,\text{thousand}\text{/}\text{southward} [/latex] in a distance of 10.0 cm. Assume that the acceleration is constant. Find the magnitude of the forcefulness exerted on the proton.
A drone is existence directed across a frictionless water ice-covered lake. The mass of the drone is i.l kg, and its velocity is [latex] 3.00\hat{i}\text{one thousand}\text{/}\text{s} [/latex]. After 10.0 s, the velocity is [latex] 9.00\hat{i}+4.00\hat{j}\text{k}\text{/}\text{s} [/latex]. If a abiding force in the horizontal direction is causing this change in motion, find (a) the components of the force and (b) the magnitude of the forcefulness.
Show Solution
a. [latex] 0.900\hat{i}+0.600\hat{j}\,\text{N} [/latex]; b. one.08 N
Source: https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/5-7-drawing-free-body-diagrams/
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