Force-triangle method for quasi-static problems

A small smooth ball of weight G rests on the flat upper face of a board that can be tilted, while a light rope keeps the ball in place. The rope is held in a FIXED direction, making a constant angle β = 37° above the horizontal and pulling the ball up the slope. Starting from a small incline angle, the board is then slowly tilted up so that its inclination θ to the horizontal increases steadily from below 37° to well above 37° (the rope direction stays fixed throughout). During this process, how do the rope tension T and the normal force N from the board on the ball change?

A worker drags a heavy crate across a level floor at constant velocity by pulling a rope that runs over a fixed pulley mounted high above. As the crate moves toward the pulley, the angle θ between the rope and the horizontal floor gradually increases from a small value toward nearly 90°. The kinetic friction coefficient between crate and floor is constant. During this process, how does the friction force f exerted by the floor on the crate change?

A small charged ball of weight G = 6.0 N hangs from a light insulating string fixed at the top. The ball carries charge and is placed in a region where it experiences an additional applied force of constant magnitude F = 3.0 N whose direction can be slowly rotated to any orientation in the vertical plane. As the direction of F is varied, the string slowly passes through a series of equilibrium positions. What is the maximum possible angle between the string and the vertical?

A small ring is held at a fixed point O on a horizontal table by two spring scales A and B. The ring also feels a constant load (a cord runs over a pulley to a fixed hanging weight), so the resultant pull that the two scales must together provide on the ring has a FIXED magnitude and a FIXED direction. Spring scale A is always kept pointing in one fixed direction. Spring scale B is rotated slowly so that the angle between B and A is gradually increased from a small acute value, through 90 degrees, to an obtuse value, while the ring stays at O. During this process, the reading of the ROTATING spring scale B:

A light horizontal rod BC is hinged to a vertical wall at end B. End C is held up by a light rope tied to a point A on the wall directly above B, so the rope makes an angle θ with the rod. A lantern of weight W hangs from end C. Keeping C fixed, a worker slowly raises point A up the wall so that the angle θ between the rope and the rod increases (θ stays between 0° and 90°). During this process, how do the tension F_T in the rope and the compression force F_N exerted by the rod on the joint at C change?

A street lamp of weight W is held in equilibrium at a fixed point B. A light rope connects B to a fixed anchor point R high on a vertical wall, so the rope direction stays fixed. A light rigid rod has its outer end fixed at B (free to pivot) and its inner end hinged to a sliding collar P on the same vertical wall; the rod acts as a compression strut. The collar P is slowly slid DOWNWARD along the wall, starting from a position only slightly below R and continuing well below B, while B, R and W stay unchanged. During this process, how does the compression force in the rod change?

A smooth (frictionless) vertical wall stands at the edge of a platform. A heavy sphere of weight W rests against the wall and is held by a light rope that passes over a small fixed pulley P mounted on the wall above the sphere. By slowly reeling in the rope at the pulley, the sphere is raised along the wall at constant (negligible) speed. As the sphere rises, the rope swings closer to the wall, so the angle between the rope and the vertical wall steadily decreases. During this slow raising, how do the rope tension T and the wall's normal force N on the sphere change?

A small lantern of weight G hangs from a light ring at point O. The ring is connected to two light ropes: rope OB is fastened to a fixed bracket so that its direction stays constant (sloping up and to the left), and rope OA is held by a worker. Keeping the lantern in equilibrium the whole time, the worker slowly moves so that rope OA rotates within the vertical plane, sweeping from a steep, nearly vertical direction down toward a more horizontal direction, passing through the position where OA is perpendicular to OB. During this process, how does the tension in rope OA change?

A small smooth ball of weight G rests against a smooth, fixed vertical wall. It is held in place by a light cord whose upper end is fixed to a point on the wall; a spring scale is inserted in the cord and reads the cord's tension. Initially the cord is nearly vertical (lying almost flat against the wall). The fixing point is then slowly moved so that the angle α between the cord and the vertical wall is gradually increased (the ball stays in equilibrium against the wall the whole time). As α increases, the reading of the spring scale

A smooth uniform cylinder of weight G rests in a symmetric V-shaped groove formed by two flat rigid plates joined at the bottom. The whole arrangement is symmetric about the vertical, and each plate makes the same angle with the vertical. Starting from a narrow groove, the two plates are slowly opened so that the opening angle between them gradually increases (the cylinder always stays wedged between the two plates and the system remains symmetric). During this slow widening, how does the magnitude of the force exerted by each plate on the cylinder change?

A smooth uniform ball of weight W rests in the corner between a fixed vertical wall and a flat board. The lower edge of the board is hinged at the base of the wall, and the board props the ball up against the wall. Starting from the horizontal position, the board is slowly rotated upward (its angle α with the horizontal is slowly increased toward 90°), keeping the ball always in contact with both the wall and the board. All surfaces are frictionless. As the board is rotated up, the normal force exerted by the BOARD on the ball

A block of mass m rests on a rough fixed inclined plane whose surface makes an angle of 30° with the horizontal. The coefficient of friction between the block and the incline is μ = 1/√3. An external force F of variable direction (its direction may be freely chosen, but it lies in the vertical plane containing the line of steepest ascent) is applied to drag the block UP the incline at constant velocity. What is the minimum magnitude of F required, and what is its direction?

A light rigid rod BC is hinged to a vertical wall at end B and points horizontally outward. A small weight of fixed magnitude W hangs from the free end C. End C is also held by a light rope AC running from C up to a point A on the wall directly above B, so that the rope makes an angle θ with the rod. By sliding point A higher up the wall, the angle θ between the rope and the rod is slowly increased from a small value toward 90° (the load W and the rod length are unchanged). During this process, how do the rope tension T and the compression force N in the rod change?

A small ball of weight mg rests on a fixed smooth inclined plane whose surface makes an angle theta = 53 deg with the horizontal. A light inextensible string lies along the incline surface (parallel to the slope) and ties the ball to a peg fixed higher up the slope, so the string can only pull the ball up the slope. In addition, a force F of constant direction acts on the ball: F is horizontal and points into the hill (toward the slope), pressing the ball against the surface. Starting from a large value, the magnitude of F is slowly decreased to zero, the ball remaining in equilibrium throughout. As F decreases, how do the string tension T and the normal force N from the incline change?

A light rigid rod AB is pinned by a smooth hinge to a vertical wall at B, and points upward and outward to its free end A. A small heavy ball of weight G hangs at A. A light rope connects A to a small ring C, and C slides slowly (quasi-statically) along a fixed smooth circular arc. During the motion the rod AB and the ball stay fixed, so the weight G keeps the same magnitude and direction and the rod's line of action stays fixed; only the rope AC rotates. As C slides, the angle between the rope AC and the rod gradually decreases, sweeping through the position where the rope is perpendicular to the rod and continuing past it. During this whole process, how do the tension in the rope and the force in the rod change?

A heavy crate of weight G hangs in the air next to a vertical wall. A light rope is tied to the crate, runs horizontally to a hook fixed on the wall, keeping the crate a fixed horizontal distance from the wall so it never touches it. A second light rope is attached to the top of the crate, passes over a fixed pulley mounted high up, and a worker slowly pulls the free end to hoist the crate straight up at constant (negligibly slow) speed. The pulley is located far above and to the side, so that the lifting rope makes an angle with the vertical that decreases steadily (the rope becomes more nearly vertical) as the crate rises. Let T1 be the tension in the lifting rope and T2 be the tension in the horizontal rope. As the crate is slowly hoisted upward, how do T1 and T2 change?

A small ball of weight W is suspended by a light string from a fixed point O on a wall. The ball rests against the smooth surface of a heavy semicylinder whose flat face stands vertically against the wall, so the curved surface bulges out and pushes the ball away from the wall. The string makes the ball lean on the cylinder. The semicylinder is now slowly slid straight DOWN along the wall (lowering it), while the suspension point O and the string length stay fixed. As the cylinder is lowered, the string becomes more nearly vertical. During this slow process, how do the tension T in the string and the normal force N exerted by the cylinder on the ball change?

A small ball of weight G is held at point O. A light horizontal string OA connects O to a vertical wall. A light rigid rod OC is hinged to the wall at C (below A) and supports the ball; the rod makes an angle theta with the horizontal. The system is in equilibrium, and theta is slowly increased from a small value toward 90 degrees. During this process, how do the tension in string OA and the force along rod OC change?

A small ball of weight G = 50 N rests on a fixed smooth (frictionless) inclined plane whose angle of inclination is θ = 37° (sin37° = 0.6, cos37° = 0.8). The ball is held in equilibrium by a light rope, and the direction of the rope can be adjusted freely (its angle relative to the incline may be changed). By changing the direction of the rope, what is the minimum possible tension in the rope, and in what direction does it then point?

A block of weight mg = 20 N rests on a rough inclined plane that makes an angle θ = 30° with the horizontal. The coefficient of friction between the block and the incline is μ = √3/3 ≈ 0.577. A person pulls the block UP the incline at constant velocity by applying a force F directed at an adjustable angle φ above the inclined surface. By choosing the optimal direction φ, what is the minimum magnitude of F required to keep the block moving up at constant velocity?

A smooth circular hoop of radius R is fixed in a vertical plane. Two small balls of weights G1 and G2 (with G1 = 2G2) rest on the inner surface of the hoop and are joined by a light rigid rod. The rod is a chord of the circle that subtends a central angle of 120 degrees. In the equilibrium position, what angle does the connecting rod make with the horizontal?

A worker drags a crate of mass m at constant velocity across a rough horizontal floor by pulling on an attached rope. The rope makes an adjustable angle θ above the horizontal, and the coefficient of kinetic friction between the crate and floor is μ = √3/3. The worker wishes to use the smallest possible pulling force F to keep the crate moving uniformly. At what angle θ should the rope be held?