Question 1

An ideal spring with a force constant (spring constant) of 15 N/m is initially compressed by 3.0 cm from its uncompressed position. How much work is required to compress the spring an additional
4)0 cm?

Accepted Answer

The work done in compressing a spring is given by the formula W = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position. In this case, the spring constant is 15 N/m and the initial displacement is 3.0 cm = 0.03 m. The additional displacement is 4.0 cm = 0.04 m. So, the work required to compress the spring an additional 4.0 cm is:W = (1/2)kx^2 = (1/2)(15 N/m)(0.04 m)^2 = 0.030 J

Question 2

It takes 87 J of work to stretch an ideal spring from 1.4 m to 2.9 m from equilibrium. What is the value of the spring constant (force constant) of this spring?

Accepted Answer

The answer of It takes 87 J of work to...

Question 3

In the figure, a ball hangs by a very light string. What is the minimum speed of the ball at the bottom of its swing (point B) in order for it to reach point A, which is 1.0 m above the bottom of the swing?

Accepted Answer

The answer of In the figure, a ball hangs by...

Question 4

A block slides down a frictionless inclined ramp and experiences no significant air resistance. If the ramp angle is 17.0° above the horizontal and the length of the surface of the ramp is 20.0 m, find
The speed of the block as it reaches the bottom of the ramp, assuming it started sliding from rest at
The top.

Accepted Answer

The speed of the block at the bottom can be found using energy conservation principles. The potential energy at the top is converted into kinetic energy at the bottom. The potential energy at the top is given by $PE = mgh$, where $h$ is the height of the ramp, and $g$ is the acceleration due to gravity (9.8 m/s$^2$). The kinetic energy at the bottom is given by $KE = \frac{1}{2}mv^2$, where $v$ is the velocity we want to find. Since $PE = KE$, we can solve for $v$.First, find the height $h$ using trigonometry: $h = L \sin(\theta)$, where $L$ is the length of the ramp (20.0 m) and $\theta$ is the angle (17.0°). So, $h = 20.0 \sin(17.0°) \approx 5.79$ m.Then, set $mgh = \frac{1}{2}mv^2$, and solve for $v$: $v = \sqrt{2gh} = \sqrt{2 \cdot 9.8 \cdot 5.79} \approx 10.7$ m/s.

Question 5

A 60-kg skier pushes off the top of a frictionless hill with an initial speed of 4.0 m/s. How fast will she be moving after dropping 10 m in elevation? Air resistance is negligible.

Accepted Answer

The answer of A 60-kg skier pushes off the top...

Question 6

How much work is required to stretch an ideal spring of spring constant (force constant)  40 N/m from  x=0.20 m to  x=0.25 m if the unstretched position is at  x=0.00 m?

Accepted Answer

The answer of How much work is required to stretch...

Question 7

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. 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 all friction and assume the spring is ideal)?

Accepted Answer

The potential energy stored in a spring is proportional to the square of the compression distance. If the spring is compressed half as far, the energy is reduced to one-fourth, so the dart will reach one-fourth of the original height, which is 6.0 meters.

Question 8

A 5.0-N projectile leaves the ground with a kinetic energy of 220 J. At the highest point in its trajectory, its kinetic energy is 120 J. To what vertical height, relative to its launch point, did it rise
If there was no air resistance?

Accepted Answer

The answer of A 5.0-N projectile leaves the ground with...

Question 9

A frictionless simple pendulum, with a small but dense 4.4-kg mass at the end and a
length of 75 cm, is released from rest at an angle of 50° with the vertical.
(a) To what height above its lowest point does the mass swing on the other side?
(b) What is the speed of the mass at the bottom of the swing?

Accepted Answer

The answer of A frictionless simple pendulum, with a small...

Question 10

A bead is moving with a speed of 20 m/s at position A on the track shown in the figure. This track is friction-free, and there is no appreciable air resistance. What is the speed of the bead at point C?

Accepted Answer

The answer of A bead is moving with a speed...

Question 11

A toy rocket that weighs 10 N blasts straight up from ground level with an initial kinetic energy of 40 J. At the exact top of its trajectory, its total mechanical energy is 140 J. To what vertical height
Above the ground does it rise, assuming no air resistance?

Accepted Answer

The total mechanical energy at the top of the trajectory is given as 140 J, which is the sum of the kinetic energy (which is 0 J at the top, since the rocket is momentarily at rest) and the potential energy. The initial kinetic energy is 40 J, and since there's no air resistance, mechanical energy is conserved. Thus, the increase in potential energy (which equals the total mechanical energy at the top minus the initial kinetic energy) is 140 J - 40 J = 100 J. The potential energy is given by $PE = mgh$, where $m$ is the mass, $g$ is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$), and $h$ is the height. The weight of the rocket is given as 10 N, which is the force due to gravity, so $m = \frac{Weight}{g} = \frac{10 \, \text{N}}{9.8 \, \text{m/s}^2}$. Solving for $h$, we get $h = \frac{PE}{mg} = \frac{100 \, \text{J}}{10 \, \text{N}} = 10 \, \text{m}$.

Question 12

A rock falls from a vertical cliff that is 4.0 m tall and experiences no significant air resistance as it falls. At what speed will its gravitational potential energy (relative to the base of the cliff) be equal
To its kinetic energy?

Accepted Answer

The answer of A rock falls from a vertical cliff...

Question 13

If a spring-operated gun can shoot a pellet to a maximum height of 100 m on Earth, how high could the pellet rise if fired on the Moon, where g = 1.6 m/s²?

Accepted Answer

The maximum height $h$ achieved by a projectile is inversely proportional to the acceleration due to gravity $g$, given by the formula $h = \frac{v^2}{2g}$, where $v$ is the initial velocity of the projectile. Since the acceleration due to gravity on the Moon ($1.6 \, \text{m/s}^2$) is $1/6$th of that on Earth ($9.8 \, \text{m/s}^2$), the height achieved on the Moon would be 6 times the height achieved on Earth, making it $100 \, \text{m} \times 6 = 600 \, \text{m}$. The closest answer to this calculation is 610 m.

Question 14

Assuming negligible friction, what spring constant (force constant) would be needed by the spring in a "B-B gun" to fire a 10-g pellet to a height of 100 m if the spring is initially compressed by 0.10
M?

Accepted Answer

To find the spring constant (k), we use energy conservation principles. The potential energy (PE) stored in the spring when compressed is equal to the gravitational potential energy (GPE) at the height of 100 m. The formula for spring potential energy is $PE_{spring} = \frac{1}{2}kx^2$, where $x$ is the compression distance (0.10 m), and for gravitational potential energy is $GPE = mgh$, where $m$ is the mass (10 g = 0.01 kg), $g$ is the acceleration due to gravity (9.8 m/s^2), and $h$ is the height (100 m). Setting these equal to each other:$\frac{1}{2}kx^2 = mgh$Solving for $k$, we get:$k = \frac{2mgh}{x^2}$Plugging in the values:$k = \frac{2 \times 0.01 \times 9.8 \times 100}{0.1^2} = 1960 \, \text{N/m}$Since the question asks for the answer in N/cm, we convert meters to centimeters by multiplying by 100 (since 1 m = 100 cm), giving us:$1960 \, \text{N/m} = 1960 \times 100 \, \text{N/cm} = 196000 \, \text{N/cm}$This calculation error led to a misunderstanding in the conversion to N/cm. Correcting this:$1960 \, \text{N/m}$ should correctly convert to $1960 \times 100 \, \text{N/cm}$ by recognizing the correct conversion factor is not applied here. The correct approach is understanding that $1 \, \text{N/m} = 0.01 \, \text{N/cm}$, thus $1960 \, \text{N/m}$ is equivalent to $19.6 \, \text{N/cm}$, which is not directly matching any of the provided options, indicating a mistake in my calculation or interpretation.Re-evaluating the calculation correctly:Given the mistake in conversion and calculation, the correct approach is to first accurately calculate the spring constant in N/m and then, if necessary, convert to N/cm with the correct conversion factor. However, the initial setup and comparison to the options provided seem to have led to confusion. The correct calculation should focus on the principles of energy conservation without conversion errors. Upon reevaluation, the mistake was in the conversion and interpretation of the final step. The correct calculation directly comparing to the options provided should yield a result that aligns with one of the given choices without necessitating incorrect conversion. The initial approach aimed to equate the potential energy in the spring to the gravitational potential energy required to reach 100 m, which is correct, but the final interpretation of the options was mishandled. Given the error in my explanation and calculation, the correct answer should be determined by accurately applying the formula and correctly interpreting the units and conversion, aiming to match the provided options without introducing conversion errors. The initial premise that energy conservation is key to solving the problem remains valid, but the execution and final interpretation need to be aligned with the correct mathematical and physical principles.

Question 15

A small but dense  2.0-kg stone is attached to one end of a very light rod that is  1.2 m long. The other end of the rod is attached to a frictionless pivot. The rod is raised until it is vertical, with the stone above the pivot. The rod is released and the stone moves in a vertical circle with no air resistance. What is the tension in the rod as the stone moves through the bottom of the circle?

Accepted Answer

The answer of A small but dense  2.0-kg stone...

Question 16

A prankster drops a water balloon from the top of a building. If the balloon is traveling at 29.1 m/s when it strikes a window ledge that is 1.5 m above the ground, how tall is the building? Neglect air
Resistance.

Accepted Answer

The answer of A prankster drops a water balloon from...

Question 17

A  1500-kg car moving at  25 m/s hits an initially uncompressed horizontal ideal spring with spring constant (force constant) of $2.0 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m }$
What is the maximum distance the spring compresses?

Accepted Answer

The maximum compression of the spring can be found by equating the initial kinetic energy of the car to the potential energy stored in the spring at maximum compression. The kinetic energy ($KE$) of the car is given by $KE = \frac{1}{2}mv^2$, where $m = 1500 \, \text{kg}$ is the mass of the car and $v = 25 \, \text{m/s}$ is its velocity. The potential energy ($PE$) stored in the spring is given by $PE = \frac{1}{2}kx^2$, where $k = 2.0 \times 10^6 \, \text{N/m}$ is the spring constant and $x$ is the compression distance.Setting $KE = PE$, we get:$$\frac{1}{2}mv^2 = \frac{1}{2}kx^2$$Solving for $x$, we find:$$x = \sqrt{\frac{mv^2}{k}} = \sqrt{\frac{(1500)(25)^2}{2.0 \times 10^6}} = \sqrt{\frac{937500}{2.0 \times 10^6}} = \sqrt{0.46875} \approx 0.68465 \, \text{m}$$Therefore, the maximum distance the spring compresses is approximately 0.68 m.

Question 18

The figure shows a famous roller coaster ride. You can ignore friction. If the roller coaster leaves point Q from rest, what is its speed at the top of the 25-m peak (point S)?

Accepted Answer

The answer of The figure shows a famous roller coaster...

Question 19

An ideal spring with a spring constant (force constant) of 22 N/mis stretched from equilibrium to 2.9 m. How much work is done in the process?

Accepted Answer

The answer of An ideal spring with a spring constant...

Question 20

What is the minimum energy needed to lift a 1.0-kg rocket to a height of 200 km and to give it a speed of 8.0 km/s at that height? (Neglect air resistance and the small decrease in g over that
Distance.)

Accepted Answer

To find the minimum energy needed, we need to calculate both the potential energy (PE) gained by lifting the rocket to a height of 200 km and the kinetic energy (KE) it has due to its speed of 8.0 km/s at that height. The potential energy gained is given by PE = mgh, where m is the mass (1.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (200,000 m, since we need to convert kilometers to meters). So, PE = 1.0 kg * 9.8 m/s^2 * 200,000 m = 1,960,000 J or 1.96 MJ.The kinetic energy is given by KE = 1/2 mv^2, where m is the mass (1.0 kg) and v is the speed (8,000 m/s, converting km/s to m/s). So, KE = 0.5 * 1.0 kg * (8,000 m/s)^2 = 32,000,000 J or 32 MJ.The total energy required is the sum of PE and KE, which is 1.96 MJ + 32 MJ = 33.96 MJ, which rounds to 34 MJ.

Quiz 6: Work and Energy

An ideal spring with a force constant (spring constant) of 15 N/m is initially compressed by 3.0 cm from its uncompressed position. How much work is required to compress the spring an additional 4) 0 cm?

It takes 87 J of work to stretch an ideal spring from 1.4 m to 2.9 m from equilibrium. What is the value of the spring constant (force constant) of this spring?

In the figure, a ball hangs by a very light string. What is the minimum speed of the ball at the bottom of its swing (point B) in order for it to reach point A, which is 1.0 m above the bottom of the swing?

A 60-kg skier pushes off the top of a frictionless hill with an initial speed of 4.0 m/s. How fast will she be moving after dropping 10 m in elevation? Air resistance is negligible.

How much work is required to stretch an ideal spring of spring constant (force constant) 40 N/m from x=0.20 m to x=0.25 m if the unstretched position is at x=0.00 m?

A 5.0-N projectile leaves the ground with a kinetic energy of 220 J. At the highest point in its trajectory, its kinetic energy is 120 J. To what vertical height, relative to its launch point, did it rise If there was no air resistance?

A bead is moving with a speed of 20 m/s at position A on the track shown in the figure. This track is friction-free, and there is no appreciable air resistance. What is the speed of the bead at point C?

A toy rocket that weighs 10 N blasts straight up from ground level with an initial kinetic energy of 40 J. At the exact top of its trajectory, its total mechanical energy is 140 J. To what vertical height Above the ground does it rise, assuming no air resistance?

A rock falls from a vertical cliff that is 4.0 m tall and experiences no significant air resistance as it falls. At what speed will its gravitational potential energy (relative to the base of the cliff) be equal To its kinetic energy?

If a spring-operated gun can shoot a pellet to a maximum height of 100 m on Earth, how high could the pellet rise if fired on the Moon, where g = 1.6 m/s²?

Assuming negligible friction, what spring constant (force constant) would be needed by the spring in a "B-B gun" to fire a 10-g pellet to a height of 100 m if the spring is initially compressed by 0.10 M?

A prankster drops a water balloon from the top of a building. If the balloon is traveling at 29.1 m/s when it strikes a window ledge that is 1.5 m above the ground, how tall is the building? Neglect air Resistance.

A 1500-kg car moving at 25 m/s hits an initially uncompressed horizontal ideal spring with spring constant (force constant) of $2.0 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m }$ What is the maximum distance the spring compresses?

The figure shows a famous roller coaster ride. You can ignore friction. If the roller coaster leaves point Q from rest, what is its speed at the top of the 25-m peak (point S) ?

An ideal spring with a spring constant (force constant) of 22 N/mis stretched from equilibrium to 2.9 m. How much work is done in the process?

What is the minimum energy needed to lift a 1.0-kg rocket to a height of 200 km and to give it a speed of 8.0 km/s at that height? (Neglect air resistance and the small decrease in g over that Distance.)

Quiz 6: Work and Energy

An ideal spring with a force constant (spring constant) of 15 N/m is initially compressed by 3.0 cm from its uncompressed position. How much work is required to compress the spring an additional 4) 0 cm?

It takes 87 J of work to stretch an ideal spring from 1.4 m to 2.9 m from equilibrium. What is the value of the spring constant (force constant) of this spring?

In the figure, a ball hangs by a very light string. What is the minimum speed of the ball at the bottom of its swing (point B) in order for it to reach point A, which is 1.0 m above the bottom of the swing?

A 60-kg skier pushes off the top of a frictionless hill with an initial speed of 4.0 m/s. How fast will she be moving after dropping 10 m in elevation? Air resistance is negligible.

How much work is required to stretch an ideal spring of spring constant (force constant) 40 N/m from x=0.20 m to x=0.25 m if the unstretched position is at x=0.00 m?

A 5.0-N projectile leaves the ground with a kinetic energy of 220 J. At the highest point in its trajectory, its kinetic energy is 120 J. To what vertical height, relative to its launch point, did it rise If there was no air resistance?

A bead is moving with a speed of 20 m/s at position A on the track shown in the figure. This track is friction-free, and there is no appreciable air resistance. What is the speed of the bead at point C?

A toy rocket that weighs 10 N blasts straight up from ground level with an initial kinetic energy of 40 J. At the exact top of its trajectory, its total mechanical energy is 140 J. To what vertical height Above the ground does it rise, assuming no air resistance?

A rock falls from a vertical cliff that is 4.0 m tall and experiences no significant air resistance as it falls. At what speed will its gravitational potential energy (relative to the base of the cliff) be equal To its kinetic energy?

If a spring-operated gun can shoot a pellet to a maximum height of 100 m on Earth, how high could the pellet rise if fired on the Moon, where g = 1.6 m/s²?

Assuming negligible friction, what spring constant (force constant) would be needed by the spring in a "B-B gun" to fire a 10-g pellet to a height of 100 m if the spring is initially compressed by 0.10 M?

A prankster drops a water balloon from the top of a building. If the balloon is traveling at 29.1 m/s when it strikes a window ledge that is 1.5 m above the ground, how tall is the building? Neglect air Resistance.

A 1500-kg car moving at 25 m/s hits an initially uncompressed horizontal ideal spring with spring constant (force constant) of 2.0×106 N/m2.0 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m }2.0×106 N/m What is the maximum distance the spring compresses?

The figure shows a famous roller coaster ride. You can ignore friction. If the roller coaster leaves point Q from rest, what is its speed at the top of the 25-m peak (point S) ?

An ideal spring with a spring constant (force constant) of 22 N/mis stretched from equilibrium to 2.9 m. How much work is done in the process?

What is the minimum energy needed to lift a 1.0-kg rocket to a height of 200 km and to give it a speed of 8.0 km/s at that height? (Neglect air resistance and the small decrease in g over that Distance.)

A 1500-kg car moving at 25 m/s hits an initially uncompressed horizontal ideal spring with spring constant (force constant) of $2.0 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m }$ What is the maximum distance the spring compresses?