Exam 7: Potential Energy and Energy Conservation

In the figure, a block of mass m is moving along the horizontal frictionless surface with a speed of 5.70 m/s. If the slope is 11.0° and the coefficient of kinetic friction between the block and the incline is 0.260, how far does the block travel up the incline?

An object of mass 4.0 kg starts at rest from the top of a rough inclined plane of height 10 m as shown in the figure. If the speed of the object at the bottom of the inclined plane is 10 m/s, how much work does friction do on this object as it slides down the incline?

Swimmers at a water park have a choice of two frictionless water slides as shown in the figure. Although both slides drop over the same height, h, slide 1 is straight while slide 2 is curved, dropping quickly at first and then leveling out. How does the speed v1 of a swimmer reaching the end of slide 1 compares with v2, the speed of a swimmer reaching the end of slide 2?

A tennis ball bounces on the floor three times. If each time it loses 22.0% of its energy due to heating, how high does it rise after the third bounce, provided we released it 2.3 m from the floor?

An 0.80-kg block is held in place against the spring by a 67-N horizontal external force (see the figure). The external force is removed, and the block is projected with a velocity v₁ = 1.2 m/s upon separation from the spring. The block descends a ramp and has a velocity v₂ = 1.9 m/s at the bottom. The track is frictionless between points A and B. The block enters a rough section at B, extending to E. The coefficient of kinetic friction over this section is 0.39. The velocity of the block is v₃ = 1.4 m/s at C. The block moves on to D, where it stops. The spring constant of the spring is closest to

In the figure, a 4.0-kg ball is on the end of a 1.6-m rope that is fixed at 0. The ball is held at point A, with the rope horizontal, and is given an initial downward velocity. The ball moves through three quarters of a circle with no friction and arrives at B, with the rope barely under tension. The initial velocity of the ball, at point A, is closest to

A 2.5-kg box, sliding on a rough horizontal surface, has a speed of 1.2 m/s when it makes contact with a spring (see the figure). The block comes to a momentary halt when the compression of the spring is 5.0 cm. The work done by the friction, from the instant the block makes contact with the spring until it comes to a momentary halt, is -0.50 J. (a) What is the spring constant of the spring? (b) What is the coefficient of kinetic friction between the box and the rough surface?

You do 174 J of work while pulling your sister back on a swing, whose chain is 5.10 m long. You start with the swing hanging vertically and pull it until the chain makes an angle of 32.0° with the vertical with your sister is at rest. What is your sister's mass, assuming negligible friction?

A 1.86-kg block is held in place against the spring by a 81-N horizontal external force (see the figure). The external force is removed, and the block is projected with a velocity v₁ = 1.2 m/s upon separation from the spring. The block descends a ramp and has a velocity v₂ = 1.9 m/s at the bottom. The track is frictionless between points A and B. The block enters a rough section at B, extending to E. The coefficient of kinetic friction over this section is 0.28. The velocity of the block is v₃ = 1.4 m/s at C. The block moves on to D, where it stops. The height h of the ramp is closest to

A 50.0-kg skier starting from rest travels 200 m down a hill that has a 20.0° slope and a uniform surface. When the skier reaches the bottom of the hill, her speed is 30.0 m/s. (a) How much work is done by friction as the skier comes down the hill? (b) What is the magnitude of the friction force if the skier travels directly down the hill?

An athlete stretches a spring an extra 40.0 cm beyond its initial length. How much energy has he transferred to the spring, if the spring constant is 52.9 N/cm?

A 2.0-kg object is moving without friction along the x-axis. The potential energy curve as a function of position is shown in the figure, and the system is conservative. If the speed of the object at the origin is 4.0 m/s, what will be its speed at 7.0 m along the +x-axis?

A spring-loaded dart gun is used to shoot a dart straight up into the air, and the dart reaches a maximum height of 24 meters above its point of release. The same dart is shot up a second time from the same gun, but this time the spring is compressed only half as far (compared to the first shot). How far up does the dart go this time? (Neglect friction and assume the spring is ideal and massless.)

Consider the motion of a 1.00-kg particle that moves with potential energy given by U(x) = (-2.00 J∙m)/x + (4.00 J∙m²)/x². Suppose the particle is moving with a speed of 3.00 m/s when it is located at x = 1.00 m. What is the speed of the object when it is located at x = 5.00 m?

A very small 100-g object is attached to one end of a massless 10-cm rod that is pivoted without friction about the opposite end. The rod is held vertical, with the object at the top, and released, allowing the rod to swing. What is the speed of the object at the instant that the rod is horizontal?

A potential energy function for system 1 is given by U1(x) = Cx² + Bx³. The potential energy function for system 2 is given by U2(x) = A + Cx² + Bx³, where A is a positive quantity. How does the force on system 1 relate to the force on system 2 at a given position?

A particle experiences a force given by F(x) = α - βx³. Find the potential field U(x) the particle is in. (Assume that the zero of potential energy is located at x = 0.)

An 8.0-m massless rod is loosely pinned to a frictionless pivot at 0, as shown in the figure. A very small 4.0-kg ball is attached to the other end of the rod. The ball is held at A, where the rod makes a 30° angle above the horizontal, and is released. The ball-rod assembly then swings freely with negligible friction in a vertical circle between A and B. The tension in the rod when the ball passes through the lowest point at D is closest to

Block 1 and block 2 have the same mass, m, and are released from the top of two inclined planes of the same height making 30° and 60° angles with the horizontal direction, respectively. If the coefficient of friction is the same in both cases, which of the blocks is going faster when it reaches the bottom of its respective incline?

A box of mass m is pressed against (but is not attached to) an ideal spring of force constant k and negligible mass, compressing the spring a distance x. After it is released, the box slides up a frictionless incline as shown in the figure and eventually stops. If we repeat this experiment but instead compress the spring a distance of 2x