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What we found:

In the figure above, the marble rolls down the track and around a loop-the-loop of radius \( R \). The marble has mass \( m \) and radius \( r \). What minimum height \( h_{min} \) must the track have for the marble to make it around the loop-the-loop without falling off? Express your answer in terms of the variables \( R \) and \( r \).

The launching mechanism of a toy gun consists of a spring with an unknown spring constant, \( k \). When the spring is compressed \( 0.120 \, \text{m} \) vertically, a \( 35.0 \, \text{g} \) projectile is able to be fired to a maximum height of \( 25 \, \text{m} \) above the position of the projectile when the spring is compressed. Assume that the barrel of the gun is frictionless.

A 1.0 kg lump of clay is sliding to the right on a frictionless surface with speed of [katex] 2 \, \{m/s} [/katex]. It collides head-on and sticks to a 0.5 kg metal sphere that is sliding to the left with speed [katex] 4\, \{m/s}[/katex]. What is the kinetic energy of the combined objects after the collision?

An object of mass 2 kg is thrown vertically downwards with an initial kinetic energy of 100 J. What is the distance fallen by the object at the instant when its kinetic energy has doubled?

A 25 g steel ball is attached to the top of a 24-cm-diameter vertical wheel. Starting from rest, the wheel accelerates at [katex] 470 \, \frac{rad}{s^2}[/katex]. The ball is released after [katex]\frac{3}{4} [/katex] of a revolution. How high does it go above the center of the wheel?

A pendulum consists of a ball of mass m suspended at the end of a massless cord of length L as shown. The pendulum is drawn aside through an angle of 60° with the vertical and released. At the low point of its swing, the speed of the pendulum ball is

A bullet moving with an initial speed of [katex] v_o [/katex] strikes and embeds itself in a block of wood which is suspended by a string, causing the bullet and block to rise to a maximum height [katex] h [/katex]. Which of the following statements is true of the collision.

While traveling in its elliptical orbit around the Sun, Mars gains speed during the part of the orbit where it is getting closer to the Sun. Which of the following can be used to explain this gain in speed?

If you want to double the momentum of a gas molecule, by what factor must you increase its kinetic energy?

In the figure, determine the character of the collision. The masses of the blocks, and the velocities before and after, are shown. The collision is

A rubber ball bounces off of a wall with an initial speed v and reverses its direction so its speed is v right after the bounce. As a result of this bounce, which of the following quantities of the ball are conserved? (There could be more than one correct choice.)

A 1.0-kg object is moving with a velocity of 6.0 m/s to the right. It collides and sticks to a 2.0-kg object moving with a velocity of 3.0 m/s in the same direction. How much kinetic energy was lost in the collision?

A force F is exerted by a broom handle on the head of a broom, which has a mass m. The handle is at an angle θ to the horizontal, as shown above. The work done by the force on the head of the broom as it moves a distance d across a horizontal floor is:

A 0.50-kg mass is attached to a spring constant 20 N/m along a horizontal, frictionless surface. The object oscillates in simple harmonic motion and has a speed of 1.5 m/s at the equilibrium position. What is the total energy of the system?

A 0.4 kg object is attached to a horizontal spring undergoes SHM with the total energy of 0.2 J. The potential energy as a function of position presented by the graph.

A 0.2 kg object is attached to a horizontal spring undergoes SHM with the total energy of 0.4 J. The kinetic energy as a function of position presented by the graph.

A small block moving with a constant speed v collides inelastically with a block M attached to one end of a spring k. The other end of the spring is connected to a stationary wall. Ignore friction between the blocks and the surface.

A 20 g piece of clay moving at a speed of 50 m/s strikes a 500 g pendulum bob at rest. The length of a string is 0.8 m. After the collision the clay-bob system starts to oscillate as a simple pendulum.

A kickball is rolled by the pitcher at a speed of 10 m/s and it is kicked by another student. The kickball deforms a little during the kick, and then rebounds with a velocity of 15 m/s as its shape restores to a perfect sphere. Select all that must be true about the kickball and the kicking foot system.

A block of mass m is moving on a horizontal frictionless surface with a speed [katex] v_0 [/katex] as it approaches a block of mass [katex] 2m [/katex] which is at rest and has an ideal spring attached to one side. When the two blocks collide, the spring is completely compressed and the two blocks momentarily move at the same speed, and then separate again, each continuing to move.

A bullet at speed v_0 hits and embeds into a wood block hanging from a string, which rises height h. Which is true about the collision?

A simple pendulum consists of a bob of mass 1.8 kg attached to a string of length 2.3 m. The pendulum is held at an angle of 30° from the vertical by a light horizontal string attached to a wall, as shown above.

The diagram above shows a marble rolling down an incline, the bottom part of which has been bent into a loop. The marble is released from point A at a height of 0.80 m above the ground. Point B is the lowest point and point C the highest point of the loop. The diameter of the loop is 0.35 m. The mass of the marble is 0.050 kg. Friction forces and any gain in kinetic energy due to the rotating of the marble can be ignored. When answering the following questions, consider the marble when it is at point C.

A boulder is raised above the ground so that its potential energy is 328 J. Then it is dropped. Assuming 90 J of energy was lost to air resistance, what is the kinetic energy of the boulder just before it hits the ground?

A 84.4 kg climber is scaling the vertical wall. His safety rope is made of a material that behaves like a spring that has a spring constant of 1.34 x 103 N/m. He accidentally slips and falls 0.627 m before the rope runs out of slack. How much is the rope stretched when it breaks his fall and momentarily brings him to rest?

A horizontal spring with spring constant 162 N/m is compressed 50 cm and used to launch a 3 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the rough surface is 0.2. Find the total distance the box travels before stopping.

If a small motor does 520 J of work to move a toy car 260 meters in a time of 37 seconds, what force does the motor exert? And how much power does the motor generate.

A big bird has a mass of about 0.021 kg. Suppose it does 0.36 J of work against gravity, so that it ascends straight up with a net acceleration of 0.625 m/s?. How far up does it move?

A 81 kg student dives off a 45 m tall bridge with an 18 m long bungee cord tied to his feet and to the bridge. You can consider the bungee cord to be a flexible spring. What spring constant must the bungee cord have for the student’s lowest point to be 2.0 m above the water?

A proton (mp = 1.67 x10-27 kg) is being accelerated along a straight line at 3.6 ×1015 m/s2 in a machine. The proton has an initial speed of 2.4 x107 m/s and travels 3.5 cm.

A mechanic pushes a 2500 kg car from rest to a final speed v by doing 5.0 x 103 J of work on the car. Frictional effect between the car and the ground are negligible. What is the final speed of the car?

A rescue helicopter lifts a 79 kg person straight up by means of a cable. The person has an upward acceleration of 0.70 m/s2 and is lifted through a distance of 11 m.

A projectile of mass 0.750 kg is shot straight up with an initial speed of 18.0 m/s.

A horizontal force of 110 N is applied to a 12 kg object, moving it 6 m on a horizontal surface where the kinetic friction coefficient is 0.25. The object then slides up a 17° inclined plane. Assuming the 110 N force is no longer acting on the incline, and the coefficient of kinetic friction there is 0.45, calculate the distance the object will slide on the incline.

A spring launches a 4 kg block across a frictionless horizontal surface. The block then ascends a 30° incline with a kinetic friction coefficient of 0.25, stopping after 55 m on the incline. If the spring constant is 800 N/m, find the initial compression of the spring. Disregard friction while in contact with the spring.

A 2 kg model rocket is launched with a thrust force of 275 N and reaches a height of 90 m, moving at 150 m/s at its peak. What is the average air resistance force acting on the rocket during its ascent?

A 5 kg cube rests against a compressed spring with a force constant of 1,800 N/m, initially compressed by 0.25 m. Upon release, the cube slides on a horizontal surface with a kinetic friction coefficient of 0.15 for 3.5 m, then ascends a 14° slope, stopping after 5 m. Determine the coefficient of kinetic friction on the slope.

It takes 4 seconds for an individual to push a 70 kg box up a 5m long, 12° ramp. The box starts from rest and achieves a speed of 2.5 m/s at the top. Friction does 350 J of work during its ascent. Calculate the power output of the individual pushing the box.

Using only work and energy, find the velocity of the masses after they have traveled 0.8 meters. Refer to the image above.

A 90 kg individual is cycling up a hill inclined at 30 degrees on a 12 kg bicycle. The hill is quite steep, and the coefficient of static friction is 0.85. The cyclist ascends 12 meters up the hill and then pauses at the summit. If they then start descending from the peak at rest and travel 9 meters before firmly applying the brakes, causing the wheels to lock.

A skier with a mass of 58 kg glides up a snowy incline that forms an angle of 28 degrees with the horizontal. The skier initially moves at a speed of 7.2 m/s. After traveling a distance of 2.3 meters up the slope, the skier’s speed reduces to 3.8 m/s.

A vehicle is moving at a speed of 12.3 m/s on a decline when the brakes of all four wheels are fully applied, causing them to lock. The slope of the decline forms an angle of 18.0 degrees with the horizontal plane. Given that the coefficient of kinetic friction between the tires and the road surface is 0.650, determine the distance the vehicle slides before it halts.

Two blocks, m2 > m1, having the same kinetic energy move from a frictionless surface onto a surface having friction coefficient [katex] \mu_k. [/katex].

A uniform solid cylinder of mass M and radius R is initially at rest on a frictionless horizontal surface. A massless string is attached to the cylinder and is wrapped around it, as shown in the figure below. The string is then pulled with a constant force F, causing the cylinder to rotate about its center of mass. After the cylinder has rotated through an angle θ, what is the kinetic energy of the cylinder (in terms of F and θ)?

A 2,000kg car collides with a stationary 1,000 kg car. Afterwards, they slide 6 m before coming to a stop. The coefficient of friction between the tires and the road is 0.7. Find the initial velocity of the 2,000kg car before the collision?

A bullet of mass 0.0500 kg traveling at 50.0 m/s is fired horizontally into a wooden block suspended from a long rope. The mass of the wooden block is 0.300 kg and it is initially at rest. The collision is completely inelastic and after impact the bullet+ wooden block move together until the center of mass of the system rises a vertical distance h above its initial position.

Block 2 initially is at rest. Block 1 travels towards block 2 and collides with Block 2 as shown above. Find the final velocities of both blocks assuming the collision is elastic.

A 0.035 kg bullet moving horizontally at 350 m/s embeds itself into an initially stationary 0.55 kg block. Air resistance is negligible.

Two boxes are tied together by a string and are sitting at rest on a frictionless surface. Between the two boxes is a massless compressed spring. The string trying the two boxes is then cut and the spring expands, pushing the boxes apart. The box on the left has four times the mass of the box on the right.

A ski lift carries skiers along a 695 meter slope inclined at 34°. To lift a single rider, it is necessary to move 72 kg of mass to the top of the lift. Under maximum load conditions, five riders per minute arrive at the top. If 65 percent of the energy supplied by the motor goes to overcoming friction, what average power must the motor supply?