1- A force F = ( 6ˆ i 2ˆ j )N acts on a particle that undergoes a displacement
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1 1- A force F = ( 6ˆ i 2ˆ j )N acts on a particle that undergoes a displacement r = ( 3ˆ i + ˆ j )m. Find (a) the work done by the force on the particle and (b) the angle between F and r. 2- The force acting on a particle varies as in the Figure. Find the work done by the force on the particle as it moves (a) from x = 0 to x = 8.00 m, (b) from x = 8.00 m to x = 10.0 m, and (c) from x = 0 to x = 10.0 m. 3- A force F = ( 4xˆ i + 3yˆ j )N acts on an object as the object moves in the x direction from the origin to x = 5.00 m. Find the work W = F dr done on the object by the force 4- A 3.00-kg object has a velocity ( 6.00ˆ i 2.00ˆ j )m /s. (a) What is its kinetic energy at this time? (b) Find the total work done on the object if its velocity changes to ( 8.00ˆ i ˆ j )m / s. 5- A crate of mass 10.0 kg is pulled up a rough incline with an initial speed of 1.50 m/s. The pulling force is 100 N parallel to the incline, which makes an angle of 20.0 with the horizontal. The coefficient of kinetic friction is 0.400, and the crate is pulled 5.00 m. (a) How much work is done by the gravitational force on the crate? (b) Determine the increase in internal energy of the crate-incline system due to friction. (c) How much work is done by the 100-N force on the crate? (d) What is the change in kinetic energy of the crate? (e) What is the speed of the crate after being pulled 5.00 m 6- A 400-N child is in a swing that is a ached to ropes 2.00 m long. Find the gravitational potential energy of the child Earth system relative to the child's lowest position when (a) the ropes are horizontal, (b) the ropes make a 30.0 angle with the vertical, and (c) the child is at the bottom of the circular arc.
2 7- A bead slides without friction around a loop-the-loop. The bead is released from a height h = 3.50R. (a) What is its speed at point A? (b) How large is the normal force on it if its mass is 5.00 g? 8- A glider of mass kg moves on a horizontal frictionless air track. It is permanently attached to one end of a massless horizontal spring, which has a force constant of 10.0 N/m both for extension and for compression. The other end of the spring is fixed. The glider is moved to compress the spring by m and then released from rest. Calculate the speed of the glider (a) at the point where it has moved m from its starting point, so that the spring is momentarily exerting no force, and (b) at the point where it has moved m from its starting point 9- A block of mass kg is placed on top of a light vertical spring of force constant 5000 N/m and pushed downward, so that the spring is compressed by m. After the block is released from rest it travels upward and then leaves the spring. To what maximum height above the point of release does it rise 10- A force acting on a particle moving in the xy plane is given by F = ( 2yˆ i + x 2ˆ j )N, where x and y are in meters. The particle moves from the origin to a final position having coordinates x = 5.00 m and y = 5.00 m, as in Figure. Calculate the work done by F along (a) OAC, (b) OBC, (c) OC. (d) Is F conservative or nonconservative? Explain
3 11- A particle of mass m = 5.00 kg is released from point A and slides on the frictionless track shown in Figure. Determine (a) the particle's speed at points B and C and (b) the net work done by the gravitational force in moving the particle from A to C. 12- A 70.0-kg diver steps off a 10.0-m tower and drops straight down into the water. If he comes to rest 5.00 m beneath the surface of the water, determine the average resistance force exerted by the water on the diver. 13- A 5.00-kg block is set into motion up an inclined plane with an initial speed of 8.00 m/s. The block comes to rest after traveling 3.00 m along the plane, which is inclined at an angle of 30.0 to the horizontal. For this motion determine (a) the change in the block's kinetic energy, (b) the change in the potential energy of the block-earth system, and (c) the friction force exerted on the block (assumed to be constant). (d) What is the coefficient of kinetic friction? 14- A boy in a wheelchair (total mass 47.0 kg) wins a race with a skateboarder. The boy has speed 1.40 m/s at the crest of a slope 2.60 m high and 12.4 m long. At the bo om of the slope his speed is 6.20 m/s. If air resistance and rolling resistance can be modeled as a constant friction force of 41.0 N, find the work he did in pushing forward on his wheels during the downhill ride.
4 15- A 50.0-kg block and 100-kg block are connected by a string as in Figure. The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between the 50-kg block and incline is Determine the change in the kinetic energy of the 50-kg block as it moves from A to B, a distance of 20.0 m ( ) N A single conservative force acting on a particle varies as F = Ax + Bx ˆi, where A and B are constants and x is in meters. (a) Calculate the potential-energy function U(x) associated with this force, taking U = 0 at x = 0. (b) Find the change in potential energy and the change in kinetic energy as the particle moves from x = 2.00 m to x = 3.00 m. 17- A potential energy function for a two-dimensional force is of the form U = 3x 3 y 7x. Find the force that acts at the point (x, y). 18- For the potential energy curve shown in Figure, (a) determine whether the force Fx is positive, negative, or zero at the five points indicated. (b) Indicate points of stable, unstable, and neutral equilibrium. (c) Sketch the curve for Fx versus x from x = 0 to x = 9.5 m.
5 19- A block slides down a curved frictionless track and then up an inclined plane as in Figure. The coefficient of kinetic friction between block and incline is µ k. Use energy methods to show that the maximum height reached by the block is y max = h 1 + µ k cotθ 20-. Suppose the incline is frictionless for the system described in the figure. The block is released from rest with the spring initially unstretched. (a) How far does it move down the incline before coming to rest? (b) What is its acceleration at its lowest point? Is the acceleration constant? (c) Describe the energy transformations that occur during the descent 21- A 10.0-kg block is released from point A in Figure. The track is frictionless except for the portion between points B and C, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant N/m, and compresses the spring m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between B and C.
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