How to Draw Inclined Plane Free Body Diagram

5 Newton'due south Laws of Motion

5.seven Drawing Free-Body Diagrams

Learning Objectives

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

• Explicate the rules for drawing a gratuitous-body diagram
• Construct free-trunk diagrams for unlike situations

The start stride in describing and analyzing nearly phenomena in physics involves the careful drawing of a gratuitous-body diagram. Costless-torso diagrams have been used in examples throughout this chapter. Remember that a costless-body diagram must only include the external forces acting on the torso of interest. Once we accept drawn an authentic free-torso diagram, we can utilize Newton's commencement 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, we gave a brief trouble-solving strategy to help you understand free-trunk diagrams. Here, we add some details to the strategy that will assistance you in constructing these diagrams.

Trouble-Solving Strategy: Constructing Gratuitous-Body Diagrams

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

1. Draw the object under consideration; it does not accept to exist creative. At first, you may want to depict a circumvolve around the object of interest to be sure yous focus on labeling the forces acting on the object. If you are treating the object as a particle (no size or shape and no rotation), represent the object every bit a indicate. Nosotros oftentimes place this betoken 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 spring force—besides equally weight and applied force. Do not include the internet force on the object. With the exception of gravity, all of the forces we have discussed require straight contact with the object. However, forces that the object exerts on its surroundings must not be included. We never include both forces of an action-reaction pair.
3. Convert the free-body diagram into a more than detailed diagram showing the x– and y-components of a given force (this is often helpful when solving a problem using Newton'due south first or 2d law). In this instance, place a squiggly line through the original vector to prove that it is no longer in play—it has been replaced past its 10– and y-components.
4. If there are two or more objects, or bodies, in the problem, draw a separate free-body diagram for each object.

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

Let'south employ the problem-solving strategy in drawing a free-body diagram for a sled. In Effigy(a), a sled is pulled by strength P at an angle of [latex]30^\circ[/latex]. In part (b), we evidence a free-body diagram for this situation, as described by steps 1 and 2 of the trouble-solving strategy. In part (c), we evidence all forces in terms of their x– and y-components, in keeping with step 3.

Effigy 5.31 (a) A moving sled is shown equally (b) a gratis-body diagram and (c) a free-torso diagram with force components.

Instance

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

Nosotros starting time by creating a diagram for the showtime object of interest. In Figure(a), object A is isolated (circled) and represented past a dot.

Figure 5.32 (a) The free-body diagram for isolated object A. (b) The costless-body diagram for isolated object B. Comparing the two drawings, nosotros see that friction acts in the opposite management in the two figures. Considering object A experiences a forcefulness 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 forcefulness must oppose it and human action up the ramp. Friction ever acts opposite the intended direction of movement.

We now include any strength 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 abroad from the interface. The source of this forcefulness is object B, and this normal strength is labeled accordingly. Since object B has a tendency to slide down, object A has a trend to slide up with respect to the interface, so the friction [latex]{f}_{\text{BA}}[/latex] is directed downward parallel to the inclined airplane.

As noted in step four of the problem-solving strategy, nosotros and then construct the gratis-body diagram in Figure(b) using the same approach. Object B experiences two normal forces and two 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 strength [latex]{N}_{\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 movement 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 first learning how to draw gratuitous-body diagrams, y'all will find it helpful to circle the object earlier deciding what forces are acting on that particular object. This focuses your attention, preventing yous from considering forces that are not acting on the body.

Example

2 Blocks in Contact

A force is applied to 2 blocks in contact, as shown.

Strategy

Draw a gratis-body diagram for each block. Be sure to consider Newton'south third police force at the interface where the 2 blocks touch.

Solution

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

Example

Block on the Table (Coupled Blocks)

A cake rests on the table, as shown. A lite rope is attached to it and runs over a pulley. The other end of the rope is attached 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 system (2 blocks and a string) to accelerate.

Strategy

Nosotros assume that the string has no mass so that nosotros practice not take to consider it as a carve up object. Draw a free-trunk diagram for each block.

Solution

Significance

Each block accelerates (notice the labels shown for [latex]{\mathbf{\overset{\to }{a}}}_{i}[/latex] and [latex]{\mathbf{\overset{\to }{a}}}_{2}[/latex]); however, bold the string remains taut, they accelerate at the same charge per unit. Thus, we take [latex]{\mathbf{\overset{\to }{a}}}_{1}={\mathbf{\overset{\to }{a}}}_{2}[/latex]. If we were to continue solving the problem, nosotros could simply telephone call the acceleration [latex]\mathbf{\overset{\to }{a}}[/latex]. Also, we apply two free-torso diagrams because we are usually finding tension T, which may require us to use a arrangement of two equations in this type of trouble. The tension is the same on both [latex]{m}_{ane}\,\text{and}\,{m}_{2}[/latex].

Check Your Understanding

(a) Depict the free-body diagram for the situation shown. (b) Redraw it showing components; use x-axes parallel to the 2 ramps.

Testify Solution

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

View this simulation to predict, qualitatively, how an external force will affect the speed and direction of an object's move. Explain the effects with the aid of a gratuitous-body diagram. Apply free-body diagrams to draw position, velocity, acceleration, 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 draw a free-trunk diagram, we describe the object of interest, depict all forces acting on that object, and resolve all force vectors into x– and y-components. We must draw a separate free-torso diagram for each object in the problem.
• A free-trunk diagram is a useful means of describing and analyzing all the forces that act on a torso to decide equilibrium according to Newton's offset law or acceleration according to Newton'south second law.

Key Equations

Net external force	[latex]{\mathbf{\overset{\to }{F}}}_{\text{net}}=\sum \mathbf{\overset{\to }{F}}={\mathbf{\overset{\to }{F}}}_{i}+{\mathbf{\overset{\to }{F}}}_{2}+\cdots[/latex]
Newton's offset police force	[latex]\mathbf{\overset{\to }{5}}=\,\text{constant when}\,{\mathbf{\overset{\to }{F}}}_{\text{cyberspace}}=\mathbf{\overset{\to }{0}}\,\text{N}[/latex]
Newton'southward 2d police, vector form	[latex]{\mathbf{\overset{\to }{F}}}_{\text{net}}=\sum \mathbf{\overset{\to }{F}}=thou\mathbf{\overset{\to }{a}}[/latex]
Newton's 2nd law, scalar class	[latex]{F}_{\text{net}}=ma[/latex]
Newton's 2d constabulary, component class	[latex]\sum {\mathbf{\overset{\to }{F}}}_{ten}=thou{\mathbf{\overset{\to }{a}}}_{x}\text{,}\,\sum {\mathbf{\overset{\to }{F}}}_{y}=m{\mathbf{\overset{\to }{a}}}_{y},\,\text{and}\,\sum {\mathbf{\overset{\to }{F}}}_{z}=m{\mathbf{\overset{\to }{a}}}_{z}.[/latex]
Newton'due south 2d constabulary, momentum form	[latex]{\mathbf{\overset{\to }{F}}}_{\text{net}}=\frac{d\mathbf{\overset{\to }{p}}}{dt}[/latex]
Definition of weight, vector grade	[latex]\mathbf{\overset{\to }{westward}}=g\mathbf{\overset{\to }{g}}[/latex]
Definition of weight, scalar class	[latex]due west=mg[/latex]
Newton's 3rd 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 grade	[latex]\mathbf{\overset{\to }{N}}=\text{−}m\mathbf{\overset{\to }{g}}[/latex]
Normal force on an object resting on a horizontal surface, scalar form	[latex]Due 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 balance, scalar form	[latex]T=w=mg[/latex]

Conceptual Questions

In completing the solution for a problem involving forces, what do we practise after constructing the complimentary-body diagram? That is, what do we use?

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

Show Solution

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

If the volume in the previous question is in free fall, how many forces should exist shown in a free-body diagram of the volume? Depict them.

Issues

A brawl of mass m 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 route. Describe a costless-body diagram; be sure to include the friction of the road that opposes the frontwards motion of the car.

Testify Solution

A runner pushes against the track, equally shown. (a) Provide a gratuitous-body diagram showing all the forces on the runner. (Hint: Place all forces at the center of his torso, and include his weight.) (b) Requite a revised diagram showing the xy-component form.

The traffic calorie-free hangs from the cables every bit shown. Draw a free-body diagram on a coordinate plane for this situation.

Prove Solution

Additional Problems

2 small forces, [latex]{\mathbf{\overset{\to }{F}}}_{i}=-2.40\mathbf{\hat{i}}-vi.10t\mathbf{\hat{j}}[/latex] North and [latex]{\mathbf{\overset{\to }{F}}}_{2}=eight.50\mathbf{\hat{i}}-nine.70\mathbf{\hat{j}}[/latex] Due north, are exerted on a rogue asteroid past a pair of space tractors. (a) Find the net force. (b) What are the magnitude and management of the internet strength? (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 North human action on an object. Their directions differ by [latex]70^\circ[/latex]. The resulting acceleration has magnitude of [latex]10.0\,{\text{m/s}}^{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^\circ[/latex]. (a) What is the dispatch of the piano up the ramp? (b) What is the velocity of the piano when information technology reaches the top if the ramp is 4.0 m long and the piano starts from rest?

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

Show 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 board 10.0 m above the water. Three seconds after entering the h2o, her down motion is stopped. What average upwards force did the water exert on her?

(a) Detect an equation to determine the magnitude of the net forcefulness required to finish a auto of mass g, given that the initial speed of the car is [latex]{v}_{0}[/latex] and the stopping distance is ten. (b) Find the magnitude of the net forcefulness if the mass of the motorcar is 1050 kg, the initial speed is 40.0 km/h, and the stopping altitude is 25.0 k.

Show Solution

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

A sailboat has a mass of [latex]1.50\times {10}^{three}[/latex] kg and is acted on by a force of [latex]ii.00\times {10}^{iii}[/latex] N toward the east, while the current of air acts backside the sails with a forcefulness of [latex]3.00\times {10}^{3}[/latex] North in a management [latex]45^\circ[/latex] northward of eastward. Find the magnitude and direction of the resulting acceleration.

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

Evidence Answer

[latex]\begin{array}{cc} {\mathbf{\overset{\to }{F}}}_{\text{internet}}=4.05\mathbf{\hat{i}}+12.0\mathbf{\hat{j}}\text{N}\hfill \\ {\mathbf{\overset{\to }{F}}}_{\text{net}}=m\mathbf{\overset{\to }{a}}\Rightarrow \mathbf{\overset{\to }{a}}=0.405\mathbf{\hat{i}}+1.20\mathbf{\hat{j}}\,{\text{m/s}}^{2}\hfill \cease{array}[/latex]

A torso of mass two.0 kg is moving along the x-axis with a speed of iii.0 thousand/s 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 subsequently 10.0 southward?

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

Prove Answer

[latex]\begin{array}{cc} {\mathbf{\overset{\to }{F}}}_{\text{internet}}={\mathbf{\overset{\to }{F}}}_{\text{A}}+{\mathbf{\overset{\to }{F}}}_{\text{B}}\hfill \\ {\mathbf{\overset{\to }{F}}}_{\text{net}}=A\mathbf{\hat{i}}+(-one.41A\mathbf{\hat{i}}-ane.41A\mathbf{\hat{j}})\hfill \\ {\mathbf{\overset{\to }{F}}}_{\text{cyberspace}}=A(-0.41\mathbf{\hat{i}}-1.41\mathbf{\lid{j}})\hfill \\ \theta =254^\circ\hfill \finish{array}[/latex]

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

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]one thousand\mathbf{\overset{\to }{1000}}[/latex]. If the trunk accelerates to the left at [latex]20\,{\text{1000/due south}}^{2}[/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 then that the speed v of the motorcar increases with position 10 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 function of position.

Show Solution

[latex]F=2kmx[/latex]; Commencement, have the derivative of the velocity function to obtain [latex]a=2kx[/latex]. Then utilise Newton's 2d police force [latex]F=ma=1000(2kx)=2kmx[/latex].

A seven.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 weather are the aforementioned but the ramp has a friction force of 1.9 N, what is the acceleration?

Two boxes, A and B, are at balance. 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 ii forces; that is, tell which one is larger or whether they are equal in magnitude. (c) If the bending of incline is [latex]ten^\circ[/latex], which force is greater?

Show Solution

a. For box A, [latex]{N}_{\text{A}}=mg[/latex] and [latex]{North}_{\text{B}}=mg\,\text{cos}\,\theta[/latex]; b. [latex]{N}_{\text{A}} \gt {N}_{\text{B}}[/latex] because for [latex]\theta \lt xc^\circ[/latex], [latex]\text{cos}\,\theta \lt i[/latex]; c. [latex]{N}_{\text{A}} \gt {N}_{\text{B}}[/latex] when [latex]\theta =10^\circ[/latex]

A mass of 250.0 k 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 grand?

As shown below, ii identical springs, each with the jump constant 20 Due north/m, support a 15.0-N weight. (a) What is the tension in jump A? (b) What is the corporeality of stretch of leap A from the rest position?

Show Solution

a. 8.66 Due north; b. 0.433 1000

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 force constant of the leap?

In building a house, carpenters use nails from a large box. The box is suspended from a bound twice during the mean solar day to measure the usage of nails. At the beginning of the day, the spring stretches 50 cm. At the end of the solar day, the leap stretches xxx cm. What fraction or percent of the nails have been used?

Testify Solution

0.40 or xl%

A force is applied to a cake to move information technology up a [latex]30^\circ[/latex] incline. The incline is frictionless. If [latex]F=65.0\,\text{Northward}[/latex] and [latex]1000=v.00\,\text{kg}[/latex], what is the magnitude of the dispatch of the block?

Two forces are applied to a 5.0-kg object, and it accelerates at a rate of [latex]two.0\,{\text{m/s}}^{2}[/latex] in the positive y-direction. If i of the forces acts in the positive x-direction with magnitude 12.0 Due north, find the magnitude of the other strength.

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

Claiming Problems

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 result of the exerted forces. (a) Describe a gratuitous-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-trunk diagram? Why or why not?

Evidence Solution

a.

b. No; [latex]{\mathbf{\overset{\to }{F}}}_{\text{R}}[/latex] is non shown, because it would replace [latex]{\mathbf{\overset{\to }{F}}}_{1}[/latex] and [latex]{\mathbf{\overset{\to }{F}}}_{2}[/latex]. (If we want to bear witness it, we could depict it and then identify squiggly lines on [latex]{\mathbf{\overset{\to }{F}}}_{ane}[/latex] and [latex]{\mathbf{\overset{\to }{F}}}_{2}[/latex] to evidence that they are no longer considered.

A x.0-kg object is initially moving east at xv.0 m/s. And then a force acts on information technology for ii.00 s, afterwards which it moves northwest, as well at fifteen.0 m/s. What are the magnitude and direction of the average force that acted on the object over the two.00-s interval?

On June 25, 1983, shot-putter Udo Beyer of East Federal republic of germany threw the seven.26-kg shot 22.22 grand, which at that fourth dimension was a world record. (a) If the shot was released at a top of 2.20 thousand with a projection angle of [latex]45.0^\circ[/latex], what was its initial velocity? (b) If while in Beyer'due south hand the shot was accelerated uniformly over a altitude of 1.20 one thousand, what was the internet force on it?

Prove Solution

a. 14.1 m/s; b. 601 Northward

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

A body of mass thousand has initial velocity [latex]{five}_{0}[/latex] in the positive ten-direction. It is acted on past a constant force F for time t until the velocity becomes zero; the force continues to act on the trunk until its velocity becomes [latex]\text{−}{v}_{0}[/latex] in the aforementioned amount of time. Write an expression for the total distance the torso travels in terms of the variables indicated.

Prove Solution

[latex]\frac{F}{grand}{t}^{2}[/latex]

The velocities of a 3.0-kg object at [latex]t=6.0\,\text{s}[/latex] and [latex]t=8.0\,\text{southward}[/latex] are [latex](iii.0\mathbf{\hat{i}}-6.0\mathbf{\hat{j}}+4.0\mathbf{\chapeau{1000}})\,\text{m/s}[/latex] and [latex](-two.0\mathbf{\hat{i}}+4.0\mathbf{\chapeau{k}})\,\text{m/s}[/latex], respectively. If the object is moving at constant acceleration, what is the force interim on it?

A 120-kg astronaut is riding in a rocket sled that is sliding along an inclined airplane. The sled has a horizontal component of acceleration of [latex]5.0\,\text{grand}\text{/}{\text{south}}^{2}[/latex] and a downward component of [latex]3.viii\,\text{thou}\text{/}{\text{due south}}^{2}[/latex]. Summate the magnitude of the force on the rider by the sled. (Hint: Remember that gravitational acceleration must exist considered.)

Two forces are acting on a 5.0-kg object that moves with dispatch [latex]ii.0\,{\text{grand/southward}}^{2}[/latex] in the positive y-direction. If one of the forces acts in the positive 10-management and has magnitude of 12 Due north, what is the magnitude of the other force?

Suppose that you lot are viewing a soccer game from a helicopter above the playing field. Two soccer players simultaneously kick a stationary soccer ball on the apartment field; the soccer ball has mass 0.420 kg. The get-go player kicks with force 162 N at [latex]9.0^\circ[/latex] north of west. At the same instant, the second actor kicks with force 215 N at [latex]15^\circ[/latex] east of southward. Notice the acceleration of the brawl in [latex]\mathbf{\lid{i}}[/latex] and [latex]\mathbf{\hat{j}}[/latex] class.

Bear witness Solution

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

A 10.0-kg mass hangs from a leap that has the bound constant 535 North/yard. Find the position of the end of the spring away from its rest position. (Use [latex]g=9.80\,{\text{m/south}}^{ii}[/latex].)

A 0.0502-kg pair of fuzzy die is attached to the rearview mirror of a motorcar past a short cord. The car accelerates at constant charge per unit, and the die hang at an angle of [latex]3.20^\circ[/latex] from the vertical because of the car'due south dispatch. What is the magnitude of the acceleration of the motorcar?

Show Solution

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

At a circus, a donkey pulls on a sled carrying a small clown with a force given by [latex]two.48\mathbf{\hat{i}}+iv.33\mathbf{\hat{j}}\,\text{Due north}[/latex]. A horse pulls on the same sled, aiding the hapless donkey, with a force of [latex]six.56\mathbf{\hat{i}}+5.33\mathbf{\hat{j}}\,\text{N}[/latex]. The mass of the sled is 575 kg. Using [latex]\mathbf{\chapeau{i}}[/latex] and [latex]\mathbf{\hat{j}}[/latex] course for the respond to each problem, find (a) the net force on the sled when the ii animals deed together, (b) the dispatch of the sled, and (c) the velocity after 6.l s.

Hanging from the ceiling over a baby bed, well out of baby'southward reach, is a cord with plastic shapes, as shown hither. The string is taut (at that place is no slack), as shown by the straight segments. Each plastic shape has the aforementioned mass thousand, and they are every bit spaced by a distance d, as shown. The angles labeled [latex]\theta[/latex] describe the angle formed past the terminate of the string and the ceiling at each end. The middle length of sting is horizontal. The remaining ii segments each form an angle with the horizontal, labeled [latex]\varphi[/latex]. Permit [latex]{T}_{ane}[/latex] be the tension in the leftmost section of the string, [latex]{T}_{2}[/latex] exist the tension in the section 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 yard, g, and [latex]\theta[/latex]. (b) Detect the angle [latex]\varphi[/latex] in terms of the angle [latex]\theta[/latex]. (c) If [latex]\theta =v.10^\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}_{ane}=\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{ane}{2}\text{tan}\,\theta )[/latex]; c. [latex]ii.56^\circ[/latex]; (d) [latex]x=d(2\,\text{cos}\,\theta +ii\,\text{cos}(\text{arctan}(\frac{1}{2}\text{tan}\,\theta ))+1)[/latex]

A bullet shot from a burglarize has mass of 10.0 g and travels to the right at 350 m/s. It strikes a target, a big purse 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 past three simultaneous forces: [latex]{\mathbf{\overset{\to }{F}}}_{i}=(-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{\hat{j}})\,\text{Northward}[/latex], and [latex]{\mathbf{\overset{\to }{F}}}_{three}=(ii.00\mathbf{\lid{i}}+five.00\mathbf{\hat{j}})\,\text{N}[/latex]. The object experiences acceleration of [latex]iv.23\,{\text{m/s}}^{ii}[/latex]. (a) Find the dispatch vector in terms of m. (b) Discover the mass of the object. (c) If the object begins from residual, find its speed after 5.00 s. (d) Find 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{3.00}{thou}\mathbf{\lid{j}})\,\text{m}\text{/}{\text{s}}^{2};[/latex] b. i.38 kg; c. 21.2 g/s; d. [latex]\mathbf{\overset{\to }{v}}=(xviii.one\mathbf{\hat{i}}+10.nine\mathbf{\hat{j}})\,\text{m}\text{/}{\text{s}}^{2}[/latex]

In a particle accelerator, a proton has mass [latex]ane.67\times {x}^{-27}\,\text{kg}[/latex] and an initial speed of [latex]two.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{m}\text{/}\text{s}[/latex] in a altitude of 10.0 cm. Presume that the dispatch is abiding. Find the magnitude of the force exerted on the proton.

A drone is existence directed across a frictionless ice-covered lake. The mass of the drone is one.50 kg, and its velocity is [latex]three.00\mathbf{\hat{i}}\text{one thousand}\text{/}\text{s}[/latex]. Subsequently 10.0 s, the velocity is [latex]nine.00\mathbf{\chapeau{i}}+4.00\mathbf{\hat{j}}\text{m}\text{/}\text{s}[/latex]. If a constant strength in the horizontal management is causing this change in motion, find (a) the components of the strength and (b) the magnitude of the force.

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a. [latex]0.900\mathbf{\lid{i}}+0.600\mathbf{\hat{j}}\,\text{Due north}[/latex]; b. 1.08 Northward

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

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