![]() ![]() The shuttle is moving to a large orbit, the r f > r i then W_ i,f 0 as expected because the component of the gravitational force is in the same direction as the displacement vector.Note that the gravitational force is in opposite direction to the radial component, therefore after doing the dot product only the radial component of the displacement vector remains. In this case, y f > y i and the W i,f > m.Īs shown in the figure, the displacement vector has two components, one is along the radial direction, the other is tangential to the circle of radius r. The result is still valid if the airplane is moving up.In this case, the gravitational force has a component in the same direction as the displacement vector and its work will make the kinetic energy increased. Because the plane is descending then y f 0 as expected.The dot product between the gravitational force and the displacement vector is: The displacement vector,, is changing along the path. What is the work done by the gravitational force to go from point to point ?Ĭonsider the coordinate system shown in the figure. The airplane starts to descend from an initial height y i, to a final height y f as shown in the figure. The control of a model airplane of mass m is not working properly. Gravity far from the Earth' surface: The gravitational force depends on position.ġ.Gravity near the Earth surface: The gravitational force is constant but the path followed by the object changes direction.We will calculate the work done by the gravitational force in the two situations: The work is positive because the force has a component in the same direction of the displacement vector.Įxample 3: Work done by the gravitational force Then the work done by the force of the wind is obtained as: The magnitude of the displacement vector is given by an element of arc length, dr=Rdθ. ![]() When the block is at an angle θ with respect to the horizontal, the displacement vector and the force form an angle 90 o - θ as shown in the figure. On the other hand, the displacement vector is tangent to the circle, therefore is changing direction changing direction. It also has a constant direction pointing towards the -x axis. In this problem the force has a constant magnitude F. The coordinate system used to descrbe the vectors is shown in the figure below. What is the work done by the wind on the block after it has moved from the bottom to the top of the hill? The force of the wind has a constant magnitude F and is parallel to the horizontal as shown in the figure. There is a constant wind hitting the block in its way up. The hill has a circular shape of radius R. The work is negative because the force is opposite to the direction of the displacement vector.Įxample 2: Changing displacement vector and a constant forceĪ block of mass m is moving up a hill. Then the magnitude of the force of kinetic friction is given by f = μ N = μ o xmg and is opposite to the direction of motion.The displacement vector has a magnitude of dx and points to the right, therefore: Because there is no motion along the vertical axis, ΣF y = 0, implies N = mg. The forces on the block along the vertical direction are the normal, N, and gravity, mg. The origin of the x-axis is where the rough region starts, therefore the work done by the force of friction on the box after it has moved a distance d is given by: ![]() What is the work done by the force of friction on the block after it has moved a distance d along the rough surface?Ĭonsider the coordinate system as shown in the figure. When either the force is not constant along the path or the displacement vector changes with position then the work is calculated as the dot product between the force and the displacement vector:Įxample 1: Moving along a straight line with a force of changing magnitudeĪ block is moving along a horizontal surface when it encounters a rough region where the coefficient of kinetic friction between the block and the surface depends on position and is given by μ(x)=μ ox, where μ o is a positive constant. Work done by a changing force or changing displacement
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