Conservation of Mechanical Energy

In summary, the problem involves a uniform solid ball rolling smoothly up an inclined ramp and stopping after a distance of 2.20 m. Using the kinematic equations and conservation of momentum, the initial speed can be calculated. However, an easier approach would be to use conservation of energy by equating the gain in gravitational potential energy to the loss in kinetic energy. Both methods result in the same answer.
  • #1
VitaX
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Homework Statement



A uniform solid ball rolls smoothly along a floor, then up a ramp inclined at 19.0 degrees. It momentarily stops when it has rolled 2.20 m along the ramp. What was its initial speed?

Homework Equations



Kinematic Equations
Conservation of Momentum

The Attempt at a Solution



I used a=(5/7)g*sin(theta)

then used V^2 - Vo^2 = 2a(x-xo) and solved for Vo. My answer is correct but I'm wondering if there is another approach to solving this problem?
 
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  • #2
You can use conservation of energy.

ehild
 
  • #3
Calculate the vertical height that the ball has gain, that's the gain in Gravitational Potential energy. There will be a equivalent lose of kinetic energy. Equate those them together solve for V.

delzac
 
  • #4
Ah ok, yes I got the same answer as I did before utilizing the rotational kinetic energy, kinetic energy and potential energy. Not as hard as I was making it out to be.
 
  • #5


I would approach this problem by first understanding the concept of conservation of mechanical energy. This principle states that in a closed system, the total amount of energy (kinetic and potential) remains constant. In this problem, the ball is the only object in motion and there are no external forces acting on it, making it a closed system.

Using this principle, we can equate the initial kinetic energy (KE) of the ball to its final potential energy (PE) at the moment it stops on the ramp. This can be represented by the equation KEi = PEf.

The initial kinetic energy of the ball can be calculated using the formula KE = 1/2mv^2, where m is the mass of the ball and v is its initial velocity. The final potential energy can be calculated using the formula PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height the ball has reached on the ramp.

Since the ball initially has no potential energy (h=0), we can write the equation as 1/2mv^2 = mgh. We can then solve for v to find the initial velocity of the ball.

Another approach to solving this problem would be to use the conservation of momentum principle. This principle states that the total momentum of a closed system remains constant. In this case, the initial momentum of the ball (mv) will be equal to its final momentum when it momentarily stops on the ramp.

Using the equation mv = mgh/tan(theta), where t is the time it takes for the ball to reach the momentary stop, we can solve for v to find the initial velocity of the ball.

Both of these approaches will lead to the same answer and it is up to the individual to choose which method they are more comfortable with. However, as a scientist, it is important to understand and apply multiple principles and equations in problem-solving to gain a deeper understanding of the concepts and to be able to verify the correctness of the answer.
 

1. What is the concept of conservation of mechanical energy?

The conservation of mechanical energy is a fundamental principle in physics that states that the total amount of mechanical energy in a closed system remains constant over time. This means that energy cannot be created or destroyed, but can only be transferred or converted from one form to another.

2. How is mechanical energy defined?

Mechanical energy is the sum of kinetic energy and potential energy of an object. Kinetic energy is the energy an object possesses due to its motion, while potential energy is the energy an object has due to its position or shape.

3. What are some examples of mechanical energy conservation in everyday life?

One example is a pendulum, where the potential energy at the highest point is converted into kinetic energy at the lowest point and then back to potential energy as the pendulum swings back and forth. Another example is a rollercoaster, where the potential energy at the top of a hill is converted into kinetic energy as the cart moves down the hill.

4. Is mechanical energy conserved in all situations?

No, mechanical energy is only conserved in closed systems where there is no external force acting on the system. In real-world scenarios, some energy is always lost due to factors like friction, air resistance, and heat.

5. How is conservation of mechanical energy applied in engineering and design?

Engineers and designers use the concept of conservation of mechanical energy to optimize the efficiency of machines and systems. By minimizing energy losses, they can create more efficient and sustainable products. Additionally, understanding this principle helps in the design of renewable energy sources, such as wind turbines and hydroelectric dams.

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