{"id":17,"date":"2023-06-27T17:03:39","date_gmt":"2023-06-27T17:03:39","guid":{"rendered":"https:\/\/physigeek.com\/nl\/machtsevenwicht\/"},"modified":"2023-06-27T17:03:39","modified_gmt":"2023-06-27T17:03:39","slug":"machtsevenwicht","status":"publish","type":"post","link":"https:\/\/physigeek.com\/nl\/machtsevenwicht\/","title":{"rendered":"Machtsevenwicht"},"content":{"rendered":"<p>In dit artikel wordt uitgelegd wat krachtbalans is en wanneer een lichaam in balans is. Ook leer je over balans van momenten en balans van krachten en momenten. Daarnaast krijg je een voorbeeld te zien en kun je oefenen met een opgeloste oefening op krachtenevenwicht.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equilibrio-de-fuerzas\"><\/span> Machtsevenwicht<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Een star lichaam verkeert in krachtevenwicht wanneer de som van alle krachten die erop worden uitgeoefend gelijk is aan nul. Met andere woorden: <strong>een lichaam is in krachtenevenwicht wanneer de resulterende kracht nul is.<\/strong><\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-ff1948e7ffb274a2dfda7b4c2bd3cb5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\somme \\vv{F}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#4fd12f\">\u27a4<\/span> <strong>Zie:<\/strong> <a href=\"https:\/\/physigeek.com\/nl\/resulterende-kracht\/\">Wat is de resulterende kracht?<\/a><\/p>\n<p> Bovendien, als een star lichaam zich in een krachtevenwicht bevindt, betekent dit dat het geen versnelling zal hebben. Daarom zal het lichaam zijn snelheid behouden of niet bewegen als het in rust is.<\/p>\n<p> Er moet rekening mee worden gehouden dat, wil een lichaam in translationeel evenwicht zijn, de som van de krachten in elke richting nul moet zijn (drie richtingen als we in de ruimte werken en twee richtingen als we in het vlak werken).<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-8173059e9d835ea8000739691bd3c498_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\somme \\vv{F_x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"53\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-f019ed19f677d908d6dc565e03921605_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\somme \\vv{F_y}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-3d5793601c780b47719e5936868096fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\somme \\vv{F_z}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Als aan \u00e9\u00e9n van de drie voorgaande voorwaarden niet wordt voldaan, zal het lichaam niet in krachtenevenwicht zijn en dus een versnelling ondergaan.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equilibrio-de-momentos\"><\/span> Tijdelijk evenwicht<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Een star lichaam verkeert in momentenevenwicht wanneer de som van alle daarop uitgeoefende momenten gelijk is aan nul. Met andere woorden, <strong>een lichaam bevindt zich in momentumevenwicht wanneer het resulterende momentum nul is.<\/strong><\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-c54de0f44f7a6549f27abfb7804a2c0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\somme \\vv{M}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> De momentenbalans is dus analoog aan de krachtenbalans, maar de som moet nul zijn in alle drie de rotatie-assen in plaats van in alle drie de lengteassen.<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-a19ea30e88dea24c5f83bcbeabe34336_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\somme \\vv{M_x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-e465040a99f8e342adb6376e49a82089_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\somme \\vv{M_y}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-2b60edfb4e54fa1744d266ff1d25c851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\somme \\vv{M_z}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Het is voldoende dat aan een eerdere vergelijking niet is voldaan, zodat de stijve vaste stof niet in momentevenwicht verkeert en daarom een rotatieversnelling heeft, of met andere woorden: het lichaam zal op zichzelf beginnen te draaien (hij begon vanuit rust) .<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equilibrio-de-fuerzas-y-momentos\"><\/span> Evenwicht van krachten en momenten<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Een star lichaam is in evenwicht van krachten en momenten wanneer de resulterende kracht en het resulterende moment nul zijn<\/strong> , dwz een lichaam is in evenwicht van krachten en momenten wanneer de som van alle krachten en alle momenten gelijk is aan nul.<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-6c79e87aef62838b71a11567cc9a620f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sum \\vv{F}=0 \\qquad \\sum\\vv{M}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Logischerwijs zal een lichaam alleen in evenwicht zijn als de som van krachten en momenten op alle assen nul is.<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-0560b8f93d7021c3c8c86531cb30833d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sum \\vv{F_x}=0\\qquad \\sum \\vv{M_x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-ea213173f7160398c238be4689cdd784_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sum \\vv{F_y}=0\\qquad \\sum \\vv{M_y}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"192\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-6f64d40d9df66dff51c3fff8b9fbad7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sum \\vv{F_z}=0\\qquad \\sum \\vv{M_z}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"191\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Zoals we hierboven zagen, hoeft een lichaam niet noodzakelijkerwijs tegelijkertijd in evenwicht te zijn tussen krachten en momenten, het kan ook alleen in krachtenevenwicht zijn en op bepaalde momenten enige onbalans vertonen, of omgekeerd.<\/p>\n<p> Wanneer een lichaam zich echter zowel in evenwicht van krachten als in evenwicht van momenten bevindt <strong>, wordt er gezegd dat het lichaam in evenwicht is<\/strong> .<\/p>\n<p> Evenwichtsomstandigheden worden gebruikt om de waarde te vinden van een kracht die op een lichaam wordt uitgeoefend, omdat ze ons in staat stellen vergelijkingen te formuleren en op basis daarvan de onbekende krachten kunnen oplossen. Normaalkracht wordt bijvoorbeeld meestal berekend door de verticale balansvergelijking op te geven.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplo-de-equilibrio-de-fuerzas\"><\/span> voorbeeld van machtsevenwicht<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Laten we, om het concept volledig te begrijpen, eens kijken naar een typisch voorbeeld van een systeem van krachten in evenwicht.<\/p>\n<p> Een op de grond gelegen lichaam is bijvoorbeeld in krachtenbalans, omdat alleen de kracht van het gewicht en de normaalkracht erop inwerken en elkaar tegenwerken. De som van de krachten en momenten in alle richtingen is dus gelijk aan nul. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/equilibre-des-forces.png\" alt=\"machtsevenwicht\" class=\"wp-image-294\" width=\"332\" height=\"264\" srcset=\"https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/equilibre-des-forces-300x238.png 300w, https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/equilibre-des-forces-768x610.png 768w, https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/equilibre-des-forces.png 852w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\"><\/figure>\n<\/div>\n<p> In dit geval verkeert het lichaam ook in evenwicht van momenten, omdat er geen moment is dat erop inwerkt. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicio-resuelto-de-equilibrio-de-fuerzas\"><\/span> Machtsevenwichtsoefening opgelost<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li> Zoals weergegeven in de volgende afbeelding zijn twee objecten verbonden door een touw en een katrol met een verwaarloosbare massa. Als object 2 een massa van 7 kg heeft en de helling van de helling 50\u00ba is, bereken dan de massa van object 1 zodat het hele systeem in evenwicht verkeert. In dit geval kan de wrijvingskracht worden verwaarloosd. <\/li>\n<\/ul>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/probleme-dequilibre-des-forces.png\" alt=\"machtsevenwichtsprobleem\" class=\"wp-image-295\" width=\"299\" height=\"240\" srcset=\"https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/probleme-dequilibre-des-forces-300x241.png 300w, https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/probleme-dequilibre-des-forces.png 718w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__FFF8E1\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#FFF8E1\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Lichaam 1 bevindt zich op een hellende helling, dus het eerste wat u moet doen is de kracht van zijn gewicht vectorisch ontbinden om de krachten in de assen van de helling te krijgen: <\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-c05811c44aa2d58295c811d612a54eee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_{1x}=P_1\\cdot \\text{sin}(\\alpha)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-1a0b77602980cc17cce9b3baef744df8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_{1y}=P_1\\cdot \\text{cos}(\\alpha)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"130\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De reeks krachten die op het hele systeem inwerken, zijn dus: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/exercice-resolu-equilibre-des-forces.png\" alt=\"machtsevenwicht is opgelost\" class=\"wp-image-296\" width=\"338\" height=\"272\" srcset=\"https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/exercice-resolu-equilibre-des-forces-300x241.png 300w, https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/exercice-resolu-equilibre-des-forces.png 718w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> De probleemstelling vertelt ons dat het krachtensysteem in evenwicht is, dus de twee lichamen moeten in evenwicht zijn. Op basis van deze informatie kunnen we de evenwichtsvergelijkingen van de twee lichamen voorstellen: <\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-ed082b4f064316ab20fb0d26054d3010_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1\\ \\rightarrow \\ \\begin{cases}P_{1x}=T\\\\[2ex]P_{1y}=N\\end{cases} \\qquad\\qquad 2 \\ \\rightarrow \\ T=P_2[\/latex ] Ainsi, la composante du poids de l'objet 1 inclin\u00e9 dans le sens de la pente doit \u00eatre \u00e9gale au poids de l'objet 2 : [latex]P_{1x}=P_2\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"1160\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-9b5e757fb28e9dde3aed458f89a3ed53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1\\cdot \\text{sen}(\\alpha)=P_2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu passen we de zwaartekrachtformule toe en vereenvoudigen we de vergelijking: <\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-06a53a846ad5bc034f69fa05488404c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_1\\cdot g \\cdot \\text{sin}(\\alpha) =m_2 \\cdot g\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"174\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-802fde26f3388538d766a709d60cf48b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_1 \\cdot \\text{sin}(\\alpha) =m_2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ten slotte vervangen we de gegevens en lossen we de massa van lichaam 1 op: <\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-0457f85ca65afde96b2e575ce54869dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_1 \\cdot \\text{sin}(50\u00ba) =7\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-9a26d132815a0ce878a6ad874c8b40b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_1 =\\cfrac{7}{\\text{sin}(50\u00ba)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"103\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-37d57b7e3c4a13f3c4dc4ae981f7d61f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_1=9,14\\kg\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In dit artikel wordt uitgelegd wat krachtbalans is en wanneer een lichaam in balans is. Ook leer je over balans van momenten en balans van krachten en momenten. Daarnaast krijg je een voorbeeld te zien en kun je oefenen met een opgeloste oefening op krachtenevenwicht. Machtsevenwicht Een star lichaam verkeert in krachtevenwicht wanneer de som &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/physigeek.com\/nl\/machtsevenwicht\/\"> <span class=\"screen-reader-text\">Machtsevenwicht<\/span> Weiterlesen &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[5],"tags":[],"class_list":["post-17","post","type-post","status-publish","format-standard","hentry","category-dynamisch"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Machtsevenwicht<\/title>\n<meta name=\"description\" content=\"Wat is krachtenevenwicht en wanneer een lichaam in evenwicht is tussen krachten en momenten. 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