brachialis antagonist

Without a proper warm-up, it is possible that you may either damage some of the muscle fibers or pull a tendon. The brachioradialis and brachialis are synergist muscles, and the rotator cuff (not shown) fixes the shoulder joint allowing the biceps brachii to exert greater force. There are also skeletal muscles that do not pull against the skeleton for movements. antagonist: infraspinatus, spinodeltoid acromi-deltoid (abducts humerous): synergist: supraspinatus antagonist: pectoralis major, latissimus dorsi spino-deltoid (extends humerous) synergist: infraspinatus Agonist muscles shorten with contraction to produce a movement. It arises from the distal part of the bone, below your biceps brachii muscle. The word oris (oris = oral) refers to the oral cavity, or the mouth. There are three subtypes of pennate muscles. For muscle pairings referred to as antagonistic pairs, one muscle is designated as the extensor muscle, which contracts to open the joint, and the flexor muscle, which acts opposite to the extensor muscle. 2023 Dotdash Media, Inc. All rights reserved, Verywell Health uses only high-quality sources, including peer-reviewed studies, to support the facts within our articles. This causes the cells in your muscle and tendon to expand and contract, heating the tissue. Read more. The Cardiovascular System: Blood Vessels and Circulation, Chapter 21. The majority of muscles are grouped in pairs, with an antagonist to each agonist muscle. We could also say that the antagonist is the main muscle that does the opposite of the action that it is resisting. Biceps brachii: in the anterior compartment of the arm, Triceps brachii: in the posterior compartment of the arm. INSERT FIGURE LIKE FOCUS FIGURE 10.1d IN MARIEB-11E. Brachialis In the shoulder elbow movement lab, this muscle is the antagonist for forearm flexion of the arm at the elbow joint. Symptoms of brachialis injury may include: People suffering from neck pain with cervical radiculopathy may experience brachialis weakness, especially if cervical level five or six is involved. Occasionally, branches from the superior and inferior ulnar collateral arteries also contribute to the arterial supply of the brachialis muscle. Compare biceps brachii, triceps brachii . The coracobrachialis is a slender muscle that originates from the deep surface of the coracoid process of scapula. This can present as a weakness when flexing the arm against resistance, but also as an inability to fully extend the elbow joint due to painful stretching of the brachialis tendon. Reading time: 4 minutes. The muscle is located medial to the biceps brachii and brachialis muscles. Register now I would honestly say that Kenhub cut my study time in half. For example, when the deltoid muscle contracts, the arm abducts (moves away from midline in the sagittal plane), but when only the anterior fascicles are stimulated, the arm will abduct and flex (move anteriorly at the shoulder joint). All of these muscles together could be referred to as synergists for flexion of the hip joint. After proper stretching and warm-up, the synovial fluid may become less viscous, allowing for better joint function. Want to learn more about terminology and the language of kinesiology? A more common name for this muscle isbelly. Each muscle fiber (cell) is covered by endomysium and the entire muscle is covered by epimysium. The attachment point for a convergent muscle could be a tendon, an aponeurosis (a flat, broad tendon), or a raphe (a very slender tendon). Wiki User. All rights reserved. Anatomy & Physiology by Lindsay M. Biga, Sierra Dawson, Amy Harwell, Robin Hopkins, Joel Kaufmann, Mike LeMaster, Philip Matern, Katie Morrison-Graham, Devon Quick & Jon Runyeon is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted. Yaw Boachie-Adjei, MD, is a board-certified, double-fellowship Orthopedic Surgeon. It has a large cross sectional area, and is able to produce more strength than the biceps brachii. Without a proper warm-up, it is possible that you may either damage some of the muscle fibers or pull a tendon. sheldonian . Pennatemuscles (penna = feathers) blend into a tendon that runs through the central region of the muscle for its whole length, somewhat like the quill of a feather with the muscle arranged similar to the feathers. During flexing of the forearm, the triceps brachii is the antagonist muscle, resisting the movement of the forearm up towards the shoulder. Massage can help decrease pain, improve blood flow, and improve tissue extensibility to the muscle. As you can see, these terms would also be reversed for the opposing action. The Spinal Cord and Spinal Nerves Ch13/14. Muscles that seem to be plump have a large mass of tissue located in the middle of the muscle, between the insertion and the origin, which is known as the central body. antagonist: infraspinatus, spinodeltoid, synergist: pectoralis major The brachialis can be clinically assessed by palpating the contracting muscle fibers during flexion of the elbow joint against resistance while the forearm is in the semi-prone position. It may be implicated in Erb's palsy if the Brachial plexus becomes injured, leading to elbow flexion weakness. For example, iliacus, psoas major, and rectus femoris all can act to flex the hip joint. When a group of muscle fibers is bundled as a unit within the whole muscle by an additional covering of a connective tissue called perimysium, that bundled group of muscle fibers is called afascicle. The first part of orbicularis, orb (orb = circular), is a reference to a round or circular structure; it may also make one think of orbit, such as the moons path around the earth. [1], The brachialis originates from the anterior surface of the distal half of the humerus,[1] near the insertion of the deltoid muscle, which it embraces by two angular processes. For example, we could say that gluteus maximus is an antagonist of the primary hip flexor, iliopsoas because gluteus maximus is a hip extensor. Kim Bengochea, Regis University, Denver. When a group of muscle fibers is bundled as a unit within the whole muscle by an additional covering of a connective tissue called perimysium, that bundled group of muscle fibers is called a fascicle. It works closely with your biceps brachii and brachioradialis muscles to ensure that your elbow bends properly. Muscles are classified according to their actions during contractions as agonists, antagonists, or synergists. antagonist: extensor digitorum, edm, synergist: fcr Following contraction, the antagonist muscle paired to the agonist muscle returns the limb to the previous position. prime mover- iliopsoas. To move the skeleton, the tension created by the contraction of the fibers in most skeletal muscles is transferred to the tendons. 2015-02-24 14:30:44. The brachialis is the major flexor of the elbow[3]. If your forearm is fully pronated, the biceps brachii is at a mechanical disadvantage, and the brachialis is the primary flexor of the elbow joint. Agonists are the prime movers while antagonists oppose or resist the movements of the agonists. When it contracts, the oral opening becomes smaller, as when puckering the lips for whistling. Grounded on academic literature and research, validated by experts, and trusted by more than 2 million users. These characteristics depend on each other and can explain the general organization of the muscular and skeletal systems. Synergist muscles act around a movable joint to produce motion similar to or in concert with agonist muscles, allowing for a range of possible movements. The muscle primarily responsible for a movement is called the prime mover, and muscles that assist in this action are called synergists. antagonist- deltoid (superior) shoulder abduction. During forearm flexion, for examplelifting a cup, a muscle called the biceps brachii is the prime mover. In a unipennate muscle, the fascicles are located on one side of the tendon. The end of the muscle that attaches to the bone being pulled is called the muscles insertion and the end of the muscle attached to a fixed, or stabilized, bone is called the origin. The effort applied to this system is the pulling or pushing on the handle to remove the nail, which is the load, or resistance to the movement of the handle in the system. In fact, nearly one-third of the students I gave the survey to was unwilling to fill it out. It has two origins (hence the "biceps" part of its name), both of which attach to the scapula bone. What is the action of the triceps brachii. The brachialis is the main muscle acting in common upper body exercises such as pull ups and elbow curls and overuse of it during exercises such as these can cause inflammation in the tendon of the muscle. C. The muscle fibers feed in on an angle to a long tendon from one side. The brachialis muscle can be commonly injured by repetitive forceful contractions or muscular contractions with the arm in hyperextension. Because of the fascicle arrangement, a portion of a multipennate muscle like the deltoid can be stimulated by the nervous system to change the direction of the pull. Triceps brachii is the antagonist and brachialis is a synergist with biceps brachii. If you suspect you have injured your brachialis muscle, visit your healthcare provider right away. When a muscle contracts, the contractile fibers shorten it to an even larger bulge. Antagonists play two important roles in muscle function: (1) they maintain body or limb position, such as holding the arm out or standing erect; and (2) they control rapid movement, as in shadow boxing without landing a punch or the ability to check the motion of a limb. When they relax, the sphincters concentrically arranged bundles of muscle fibers increase the size of the opening, and when they contract, the size of the opening shrinks to the point of closure. There also are skeletal muscles in the tongue, and the external urinary and anal sphincters that allow for voluntary regulation of urination and defecation, respectively. With less pain, you may be able to fully engage in your rehab program for your injured brachialis. It lies beneath the biceps brachii, and makes up part of the floor of the region known as the cubital fossa (elbow pit). These terms arereversed for the opposite action, flexion of the leg at the knee. Fascicles can be parallel, circular, convergent, or pennate. Author: antagonist: ecrl, ecrb, ecu, synergist: fds, fdp Rewrite it, correcting all errors. The coracobrachialis muscle is innervated by the musculocutaneous nerve (C5-C7) a branch of the lateral cord of the brachial plexus. brachialis, brachioradialis. St. Louis, MO: Mosby/Elsevier; 2011. acts as the antagonist. The brachialis is the only pure flexor of the elbow jointproducing the majority of force during elbow flexion. (credit: Victoria Garcia). One of our most important requirements are good role models. Lever systems in the human body are classified based on the arrangement pattern of the fulcrum, resistance, and the applied force (Figure \(\PageIndex{4}\)). and grab your free ultimate anatomy study guide! Antagonists . Curated learning paths created by our anatomy experts, 1000s of high quality anatomy illustrations and articles. The coracobrachialis does flexion and adduction of the arm at the shoulder. There are four helpful rules that can be applied to all major joints except the ankle and knee because the lower extremity is rotated during development. They are thus antagonist muscles. Along with the humerus, coracobrachialis forms the lateral border of the axilla, where it is also the easiest to palpate the muscle. It contains four muscles - three in the anterior compartment (biceps brachii, brachialis, coracobrachialis), and one in the posterior compartment (triceps brachii). To pull on a bone, that is, to change the angle at its synovial joint, which essentially moves the skeleton, a skeletal muscle must also be attached to a fixed part of the skeleton. A bipennate muscle has fascicles on both sides of the tendon, as seen in rectus femoris of the upper leg. https://rad.washington.edu/muscle-atlas/brachialis/, Distal insertional footprint of the brachialis muscle: 3D morphometric study. Which of the following helps an agonist work? Federative Committee on Anatomical Terminology (FCAT) (1998). What follows are the most common fascicle arrangements. When the fulcrum lies between the resistance and the applied force, it is considered to be a first class lever (Figure \(\PageIndex{4.a}\)). The Cardiovascular System: The Heart, Chapter 20. What actions does the coracobrachialis muscle do? Skeletal muscles each have an origin and an insertion. Muscles are arranged in groupings of agonist, antagonist, and synergists that produce and modulate movement. The majority of skeletal muscles in the body have this type of organization. Parallel muscles have fascicles that are arranged in the same direction as the long axis of the muscle. Also known as the overhand curl, this brachioradialis exercise directly targets your forearms and biceps. Each muscle fiber (cell) is covered by endomysium and the entire muscle is covered by epimysium. Write TRT RTR for transitive or int. prime mover- deltoid (superior) synergist- supraspinatus. Laterally it is related to the radial nerve in the radial groove, the brachioradialis and the extensor carpi radialis longus muscles. For example, in the case of the knee, muscles of the posterior thigh cause knee flexion and anterior thigh muscles cause knee extension, which is opposite of the rules stated below for most other joints. A muscle that has a pattern of fascicles running along the long axis of the muscle has which of the following fascicle arrangements? It is a major flexor of the forearm at the elbow joint, flexing the elbow while it is in all positions. Read more. Kim Bengochea, Regis University, Denver. "Brachialis Muscle." There are other muscles throughout the body named by their shape or location. The information we provide is grounded on academic literature and peer-reviewed research. The information we provide is grounded on academic literature and peer-reviewed research. A typical symptom is pain in the arm and shoulder, radiating down to the back of the hand. Our musculoskeletal system works in a similar manner, with bones being stiff levers and the articular endings of the bonesencased in synovial jointsacting as fulcrums. The muscles of the rotator cuff are also synergists in that they fix the shoulder joint allowing the bicepps brachii to exert a greater force. Standring, S. (2016). This gradually increases with the regaining of strength in the muscle. Some parallel muscles are flat sheets that expand at the ends to make broad attachments. Pennate muscles (penna = feathers) blend into a tendon that runs through the central region of the muscle for its whole length, somewhat like the quill of a feather with the muscle arranged similar to the feathers. For example, there are the muscles that produce facial expressions. [9], Get Top Tips Tuesday and The Latest Physiopedia updates, The content on or accessible through Physiopedia is for informational purposes only. In this motion the atlanto-occipital joint is the fulcrum, the head is the resistance, and the applied force is generated by the trapezius muscle of the neck. However, to flex the knee joint, an opposite or antagonistic set of muscles called the hamstrings is activated. Everyone need to look up to somebody. Bone Tissue and the Skeletal System, Chapter 12. For example, when the deltoid muscle contracts, the arm abducts (moves away from midline in the sagittal plane), but when only the anterior fascicle is stimulated, the arm willabductand flex (move anteriorly at the shoulder joint). If you are experiencing pain in the front of your elbow due to a brachialis injury, you may benefit from using electrical stimulation to the area. Most of the joints you use during exercise are synovial joints, which have synovial fluid in the joint space between two bones. Circular muscles are also called sphincters (Figure \(\PageIndex{2}\)). temporalis (elevates mandible) : antagonist? The effort applied to this system is the pulling or pushing on the handle to remove the nail, which is the load, or resistance to the movement of the handle in the system. These characteristics depend on each other and can explain the general organization of the muscular and skeletal systems. The brachialis ( brachialis anticus ), also known as the Teichmann muscle, is a muscle in the upper arm that flexes the elbow. The Peripheral Nervous System, Chapter 18. D. The muscle fibers on one side of a tendon feed into it at a certain angle and muscle fibers on the other side of the tendon feed into it at the opposite angle. extensor muscles during instructed flexions: fixator: supraspinatus, infraspinatus, teres minor and subscapularis muscles: The main flexor of the elbow is the brachialis muscle. The prime mover, sometimes called the agonist, is the muscle that provides the primary force driving the action. antagonist: triceps brachii, synergist: brachioradialis, biceps brachii Anatomy and human movement: structure and function (6th ed.). Turn your forearm over into a pronated position, and have someone press down, attempting to straighten your elbow. Hamstrings: group of three muscles in the posterior compartment of the thigh, Quadriceps femoris: group of four muscles in the anterior compartment of the thigh. While we need the main muscle, or agonist, that does an action, our body has a good support system for each action by using muscle synergists. This is called brachialis tendonitis. This page titled 10.2: Interactions of Skeletal Muscles, Their Fascicle Arrangement, and Their Lever Systems is shared under a CC BY license and was authored, remixed, and/or curated by Whitney Menefee, Julie Jenks, Chiara Mazzasette, & Kim-Leiloni Nguyen (ASCCC Open Educational Resources Initiative) . The hamstrings flex the leg, whereas the quadriceps femoris extend it. The brachialis is the only pure flexor of the elbow joint-producing the majority of force during elbow flexion. The brachialis is the major flexor of the elbow [3]. The temporalis muscle of the cranium is another. Meanwhile, a muscle with the opposite action of the prime mover is called an antagonist. Skeletal muscle is enclosed in connective tissue scaffolding at three levels. 2023 If you consider the first action as the knee bending, the hamstrings would be called the agonists and the quadriceps femoris would then be called the antagonists. Agonist muscles are those we typically associate with movement itself, and are thus sometimes referred to as prime movers. Although we learn the actions of individual muscles, in real movement, no muscle works alone. Learn everything about the anatomy of the shoulder muscles with our study unit. Many of us doesn't seem to look up to anybody at all. It works closely with your biceps brachii and brachioradialis muscles to ensure that your elbow bends properly. tricepts brachii when the brachialis acts as the agonist, the __ muscle assists as a synergist antagonist a ___ is a synergistic muscle that will stabilize a joint when another contracting muscle exerts a force on something else . Kenhub. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Flexing of the forearm by the biceps brachii: The biceps brachii is the agonist, or primer mover, responsible for flexing the forearm. The flexor digitorum superficialis and flexor digitorum profundus flex the fingers and the hand at the wrist, whereas the extensor digitorum extends the fingers and the hand at the wrist. It can also fixate the elbow joint when the forearm and hand are used for fine movements, e.g., when writing. The brachoradialis, in the forearm, and brachialis, located deep to the biceps in the upper arm, are both synergists that aid in this motion. The heads of the muscle arise from the scapula (shoulder blade) and . A synergist can also be a fixator that stabilizes the bone that is the attachment for the prime movers origin. synergist and antagonist muscles. Triceps brachii In the Shoulder elbow movement lab, this muscle is the prime mover for abduction of the arm at the shoulder joint. The the body (resistance), lies between the metatarsophalangeal joints (fulcrum), and the applied forced from several lower leg muscles including gastrocnemius. I cracked my wristwatch against the doorpost this morning on my way out the door. Synergists assist the agonists, and fixators stabilize a muscles origin. By understanding the anatomy and function of the brachialis muscle, you can be sure to have a successful rehab process and quickly and safely return to your previous level of activity. The Chemical Level of Organization, Chapter 3. During flexing of the forearm, the brachioradialis and brachialis act as synergist muscles, aiding the biceps brachii in pulling the forearm up towards the shoulder. A muscle that crosses the anterior side of a joint results in flexion, which results in a decrease in joint angle with movement. Our website is not intended to be a substitute for professional medical advice, diagnosis, or treatment. Philadelphia, PA: Lippincott Williams & Wilkins. Muscles that seem to be plump have a large mass of tissue located in the middle of the muscle, between the insertion and the origin, which is known as the central body, or belly. antagonist: infraspinatus, spinodeltoid, synergist: supraspinatus Hamstrings: group of three muscles in the posterior compartment of the thigh, Quadriceps femoris: group of four muscles in the anterior compartment of the thigh. A muscle with the opposite action of the prime mover is called an antagonist. Prevention of injuries to muscles can be achieved by correctly warming up before exercise, but may also include the use of external accessories such as bandages and tapes. About a week after your injury, heat may be applied to improve circulation to the muscle and to allow it to stretch a little more easily. By Brett Sears, PT This answer is: Study guides. A synergist that makes the insertion site more stable is called a fixator. There are also skeletal muscles that do not pull against the skeleton for movements. Rear Front Rotations. It simply heats the tissue. It functions to flex the forearm. Skeletal muscle is enclosed in connective tissue scaffolding at three levels. Other parallel muscles are rotund with tendons at one or both ends. Distal anterior aspect of the humerus, deep to the biceps brachii. The extensor digitorum of the forearm is an example of a unipennate muscle. The brachialis (brachialis anticus), also known as the Teichmann muscle, is a muscle in the upper arm that flexes the elbow. Access for free athttps://openstax.org/books/anatomy-and-physiology. https://en.wikipedia.org/w/index.php?title=Brachialis_muscle&oldid=1133471006, This page was last edited on 13 January 2023, at 23:54. Palastanga, N., & Soames, R. (2012). B. The humeral insertion of coracobrachialis is crossed anteriorly by the median nerve. What effect does fascicle arrangement have on a muscles action? Fascicle arrangement by perimysia is correlated to the force generated by a muscle; it also affects the range of motion of the muscle. Edinburgh: Elsevier Churchill Livingstone. INSERT FIGURE LIKE FOCUS FIGURE 10.1c IN MARIEB-11E. The biceps brachii flexes the forearm, whereas the triceps brachii extends it. Moore, K. L., Dalley, A. F., & Agur, A. M. R. (2014). Balance between a muscle agonist, its synergists, and its antagonist(s) is important for healthy movement and avoiding pain and injury. It has been suggested that the long head fascicle is employed when sustained force generation is demanded, or when there is a . The brachialis is a muscle located in your arm near the crook of your elbow. Brett Sears, PT, MDT, is a physical therapist with over 20 years of experience in orthopedic and hospital-based therapy. A second class lever is arranged with the resistance between the fulcrum and the applied force (Figure \(\PageIndex{4.b}\)). Made with by Yoganatomy.com and Wildheartmedia.com. For example, the biceps brachii functions to produce the movement of elbow flexion. Coracobrachialis muscle: want to learn more about it? Q. [cited 2018 Mar 21]. and What Is Muscle Origin, Insertion, and Action? The Nervous System and Nervous Tissue, Chapter 13. Neither books nor just thinking about it help as much as walking in a hero's footsteps. It is not affected by pronation or supination of the forearm, and does not participate in pronation and supination due to its lack of attachment to the radius. Massage may also be an effective treatment for brachialis injury. synergist- Sartorius, rectus femoris, gracilis, tensor fasciae late. When you visit the site, Dotdash Meredith and its partners may store or retrieve information on your browser, mostly in the form of cookies. Cookies collect information about your preferences and your devices and are used to make the site work as you expect it to, to understand how you interact with the site, and to show advertisements that are targeted to your interests. The brachialis is known as the workhorse of the elbow. The triceps is an extensor muscle of the elbow joint and an antagonist of the biceps and brachialis muscles. Reviewer: An Introduction to the Human Body, Chapter 2. In the horse, the brachial muscle ends with . Read more. It is often performed prior to stretching. When it contracts, the oral opening becomes smaller, as when puckering the lips for whistling. Describe how muscles are arranged around the joints of the body. The first part of orbicularis, orb (orb = circular), is a reference to a round or circular structure; it may also make one think of orbit, such as the moons path around the earth. The insertions and origins of facial muscles are in the skin, so that certain individual muscles contract to form a smile or frown, form sounds or words, and raise the eyebrows. The brachialis is known as the workhorse of the elbow. Q. 2013 [cited 2018 Mar 21]. This is commonly seen in climbers, due to the pronation of the hand and the extended started position. ), Brachialis muscle (labeled in green text), This article incorporates text in the public domain from page 444 ofthe 20th edition of Gray's Anatomy (1918), Deep muscles of the chest and front of the arm, with the boundaries of the. Due to this design, the muscle fibers in a pennate muscle can only pull at an angle, and as a result, contracting pennate muscles do not move their tendons very far. In most cases Physiopedia articles are a secondary source and so should not be used as references. If you are able to withstand the force of someone pushing your arm, then brachialis strength is considered to be adequate. The humerus and the capsule of the elbow joint lie posterior to the muscle. Agonist muscles produce the primary movement or series of movements through their own contractions. This corresponds to a spinal level of cervical five and six. Explain how a synergist assists an agonist by being a fixator. All rights reserved. Kenhub, Aug. 2001, "Brachialis." The triceps brachii muscle is located on the back of the arm and, when contracts, straightens the elbow joint. When you first get up and start moving, your joints feel stiff for a number of reasons. Another example is the orbicularis oculi, one of which surrounds each eye. Author: A synergist can also be a fixator that stabilizes the muscles origin. [5] In order to isolate the brachialis muscle the forearm needs to be in pronation, due to the biceps brachii's function as a supinator and flexor. Start now! Boston, Ma: Pearson; 2016. Another example is the orbicularis oculi, one of which surrounds each eye. Verywell Health's content is for informational and educational purposes only. Anatomy, Shoulder and Upper Limb, Brachialis Muscle. The tendons of the bicep connect to the upper arm and the forearm. The biceps brachii has two synergist muscles that assist it in flexing the forearm. Available from: Kamineni S, Bachoura A, Behrens W, Kamineni E, Deane A. Marieb EN, Hoehn K. 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"18:_Cardiovascular_System_-_Blood_Vessels_and_Circulation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "19:_Lymphatic_and_Immune_System" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "20:_Respiratory_System" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "21:_Digestive_System" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "22:_Urinary_System" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "23:_Reproductive_System" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, 10.2: Interactions of Skeletal Muscles, 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\newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Reedley College, Butte College, Pasadena City College, & Mt.

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