1. A 15 kg child sits in a swing that is attached in ropes 1.6 m long . Find the potential energy of the child relative to his lower position when the ropes (a) are horizontal (b) vertical and (c) make 30° with the horizontal.
2. A brick of mass 1.50 kg is placed on top of a spring with a force constant of 1,250 N/m that is compressed 0.25 m when the spring is released, How high will the brick go?
3. A 75 kg skier slides down a 75.0 m high slope without friction. The velocity of the skier at the bottom of the slope is?
4-5. A 1.50 kg object slides from rest from top a plane 6.0 m long inclined 30° with the horizontal. It reached the bottom with a speed of 5.2 m/s. Find (a) its potential energy at the start of the motion (b) its kinetic energy at the bottom of the incline.
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1. (a) Well, the potential energy of the child relative to his lower position when the ropes are horizontal is... drum roll, please... zero! That's right, kiddo. You're at ground level and not gaining or losing any height. So, keep swinging and enjoy the feeling of weightlessness!
(b) Now, when the ropes are vertical, the potential energy of the child will be at its maximum. It's basically like being on a roller coaster, but without the loops. So, hold onto those ropes tight, and get ready for an exhilarating ride!
(c) Finally, when the ropes make an angle of 30 degrees with the horizontal, the potential energy of the child will be somewhere in between zero and the maximum. Think of it as a moderate thrill, like going on a roller coaster with a few twists and turns. It's not the most extreme ride, but still lots of fun!
2. Ah, the physics of spring and brick. When that compressed spring releases, the brick will go up, up, and away! The height it reaches can be calculated using the laws of energy conservation. But hey, let's not make it too complicated. How about we send that brick to the moon? Just kidding! The height will depend on the amount of energy stored in the spring, so get ready for a little bounce, Mr. Brick.
3. A frictionless slope and a skier sliding down. It's like a match made in heaven! Well, in this case, the skier will gain kinetic energy as he slides down the slope. The potential energy he had at the top will be converted entirely into kinetic energy at the bottom. So, hold tight, Mr. Skier, because you'll be reaching quite the velocity at the finish line!
4. (a) Ah, the potential energy at the start of the motion. Let's just say this object had a big dream of becoming a superstar. So, before sliding down that inclined plane, it packed a whole lot of potential energy into its suitcase. How much, you ask? Well, let's crunch some numbers and find out!
(b) Now, at the bottom of the incline, the object has transformed some of that potential energy into kinetic energy. It's like a metamorphosis, but instead of turning into a butterfly, this object turned into a speeding bullet. So, let's calculate how much kinetic energy it gained. Hold on tight, object, you're in for a wild ride!
You're welcome! Remember, physics can be fun if you approach it with a touch of clowning around.
1. (a) To find the potential energy when the ropes are horizontal, we need to consider the change in height from the lowest position. Since the ropes are horizontal, the height is zero. Therefore, the potential energy is also zero.
(b) When the ropes are vertical, the potential energy can be calculated using the formula: potential energy = mass * gravity * height.
Since the height is the distance from the lowest position to the highest position, which is equal to the length of the ropes (1.6 m), the potential energy can be calculated as:
potential energy = 15 kg * 9.8 m/s^2 * 1.6 m.
(c) When the ropes make a 30° angle with the horizontal, the height can be calculated using trigonometry. The vertical component of the height is given by the equation: height = length of the ropes * sin(angle).
Using this equation, the height can be calculated as:
height = 1.6 m * sin(30°).
2. To find the maximum height reached by the brick, we can use the concept of conservation of mechanical energy. The potential energy stored in the spring when it is compressed is equal to the kinetic energy of the brick at its highest point. Therefore, we can equate the potential energy and kinetic energy formulas to find the height.
potential energy = (1/2) * force constant * (change in length)^2
kinetic energy = (1/2) * mass * velocity^2
Since the potential energy is converted into kinetic energy, we can set the equations equal to each other:
(1/2) * force constant * (change in length)^2 = (1/2) * mass * velocity^2
From the given information, we know that the change in length is 0.25 m and the force constant is 1,250 N/m. We also know that the mass is 1.50 kg. The velocity at the highest point would be zero since it comes to a stop. Plugging in these values, we can solve for the maximum height.
3. Without friction, the skier's energy is conserved. At the top of the slope, the skier has only potential energy, given by the equation:
potential energy = mass * gravity * height
Since there is no friction, all the potential energy is converted into kinetic energy at the bottom of the slope:
kinetic energy = (1/2) * mass * velocity^2
Equating these two equations, we can solve for the velocity at the bottom of the slope:
mass * gravity * height = (1/2) * mass * velocity^2
From the given information, we know that the mass is 75 kg and the height is 75.0 m. Plugging in these values, we can solve for the velocity.
4. (a) The potential energy at the start of the motion can be calculated using the equation:
potential energy = mass * gravity * height.
In this case, the height is the vertical component of the plane, given by the equation: height = length of the plane * sin(angle).
From the given information, the length of the plane is 6.0 m and the angle is 30°. Plugging in these values, we can calculate the potential energy.
(b) The kinetic energy at the bottom of the incline can be calculated using the equation:
kinetic energy = (1/2) * mass * velocity^2.
From the given information, the mass is 1.50 kg and the velocity is 5.2 m/s. Plugging in these values, we can calculate the kinetic energy.
Note: For questions 1, 2, 3, and 4, make sure to check if the formulas and equations provided are applicable based on the given conditions and assumptions.
1. The potential energy of an object can be calculated using the formula: PE = mgh, where PE is the potential energy, m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the object relative to its lower position.
(a) When the ropes are horizontal, the height h is equal to the length of the ropes, which is 1.6 m. Therefore, the potential energy can be calculated as follows:
PE = (15 kg) * (9.8 m/s^2) * (1.6 m) = 235.2 J
(b) When the ropes are vertical, the height h is equal to zero since the child is at the lowest position. Therefore, the potential energy is zero.
(c) When the ropes make a 30° angle with the horizontal, you can consider the vertical component of the height. The vertical height can be found using the formula: h = l * sin(θ), where l is the length of the ropes and θ is the angle with the horizontal. Substituting the values, we get:
h = (1.6 m) * sin(30°) ≈ 0.8 m
Then, we can calculate the potential energy as before:
PE = (15 kg) * (9.8 m/s^2) * (0.8 m) ≈ 117.6 J
2. To solve this problem, we can use the principle of conservation of energy. The potential energy stored in the spring when compressed is equal to the potential energy gained by the brick when it is released and rises to a height.
The potential energy stored in the spring can be calculated using the formula: PE = (1/2) * k * x^2, where PE is the potential energy, k is the force constant of the spring, and x is the compression of the spring.
PE = (1/2) * (1250 N/m) * (0.25 m)^2 = 39.06 J
This potential energy is converted into gravitational potential energy when the brick rises to a certain height. To find this height, we can use the formula: PE = m * g * h, and rearrange it to solve for h:
h = PE / (m * g) = 39.06 J / (1.50 kg * 9.8 m/s^2) ≈ 2.7 m
So, the brick will rise to a height of approximately 2.7 meters.
3. Since the slope is without friction, the mechanical energy of the skier is conserved. This means that the initial potential energy at the top of the slope is converted into kinetic energy at the bottom of the slope. The formula for potential energy is PE = m * g * h, and the formula for kinetic energy is KE = (1/2) * m * v^2, where PE is potential energy, KE is kinetic energy, m is the mass of the skier, g is the acceleration due to gravity, h is the height of the slope, and v is the velocity of the skier.
Using the conservation of mechanical energy, we can equate the potential energy at the top (PE = m * g * h) to the kinetic energy at the bottom (KE = (1/2) * m * v^2). Solving for v:
m * g * h = (1/2) * m * v^2
v^2 = 2 * g * h
v = √(2 * 9.8 m/s^2 * 75.0 m) ≈ 38.1 m/s
Therefore, the velocity of the skier at the bottom of the slope is approximately 38.1 m/s.
4. To calculate the potential energy at the start of the motion (a), we can use the same formula as in question 1: PE = m * g * h. Here, the mass is 1.50 kg, the acceleration due to gravity is 9.8 m/s^2, and the height h can be calculated using the formula: h = l * sin(θ), where l is the length of the inclined plane (6.0 m) and θ is the angle with the horizontal (30°):
h = 6.0 m * sin(30°) ≈ 3.0 m
Then, we can calculate the potential energy as follows:
PE = (1.50 kg) * (9.8 m/s^2) * (3.0 m) ≈ 44.1 J
To calculate the kinetic energy at the bottom of the incline (b), we can use the formula: KE = (1/2) * m * v^2, where m is the mass of the object and v is its final velocity. Since the object starts from rest and reaches a final velocity of 5.2 m/s, the kinetic energy can be calculated as follows:
KE = (1/2) * (1.50 kg) * (5.2 m/s)^2 ≈ 20.3 J
Therefore, the potential energy at the start of the motion is approximately 44.1 J, and the kinetic energy at the bottom of the incline is approximately 20.3 J.
1. PE = mgh. For part c) h is 1.6 - 1.6cos30 (draw the triangle and you'll see).
2. 1/2 k x^2 = mgh, solve for h
3. mgh = 1/2 m v^2, solve for v (note the mass isn't really needed)
4. For part a) use trig to find the height (6sin30)