Conservation of mechanical energy – problems and solutions

1. An m-kg block is released from the top of the smooth inclined plane, as shown in the figure below. Comparison between the gravitational potential energy and kinetic energy of the block at point M is…

__Solution__

Gravitational potential energy at point M :

PE_{M} = m g (1/3h) = 1/3 m g h

Kinetic energy at point M :

KE_{M} = EP = m g (2/3 h) = 2/3 m g h

EP_{M} : EK_{M }

1/3 m g h : 2/3 m g h

1/3 : 2/3

1 : 2

2. An ice skier sliding from a height of A, as shown in the figure below. If the initial velocity = 0 and acceleration due to gravity is 10 ms^{-2}, then what is the velocity of the skier at point B.

__Known :__

Initial velocity (v_{o}) = 0

Acceleration due to gravity (g) = 10 m/s^{2}

The change in height = 50 meters – 10 meters = 40 meters

__Wanted:__ Velocity of the skier at point B

__Solution :__

The principle of conservation of mechanical of energy states that the initial mechanical energy = the final mechanical energy.

The initial mechanical energy = potential energy at point B (height = 40 meters)

The final mechanical energy = kinetic energy at point B

The final mechanical energy = The initial mechanical energy

½ m v_{t}^{2 }= m g h

½ v_{t}^{2 }= g h

½ v_{t}^{2} = (10)(50-10)

½ v_{t}^{2 }= (10)(40)

½ v_{t}^{2 }= 400

v_{t}^{2 }= (2)(400) = 800

v_{t} = √800 = √(2)(400) = 20√2 m/s

3. An object start to sliding from a point of A without the initial velocity. If there is no friction force, what is the velocity of the object at the lowest point.

__Known :__

Mass of object = m

Initial velocity (v_{o}) = 0

Height (h) = 20 meters

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted :__ Final velocity (v_{t})

__Solution :__

Initial mechanical energy (ME_{1}) = Gravitational potential energy at point A (PE_{A}) = m g h = (m)(10)(20) = 200 m

Final mechanical energy (ME_{2}) = kinetic energy (KE) = ½ m v_{t}^{2}

__The velocity of the object at the lowest point ____(v___{t}__) ?__

Apply the principle of conservation of mechanical of energy states that the initial mechanical energy = the final mechanical energy.

EM1 = EM_{2}

200 m = ½ m v_{t}^{2}

200 = ½ v_{t}^{2}

400 = v_{t}^{2}

v_{t} = 20 m/s

4. A 2-kg ball free fall from point A, as shown in figure below (g = 10 ms^{-2}). After arrive at point B, the kinetic energy = 2 times the potential energy. What is the height of point B above the surface of earth.

__Known :__

Mass of ball (m) = 2 kg

Acceleration due to gravity (g) = 10 ms^{-2}

Height of point A (h_{A}) = 90 meters

__Wanted:__ Height of point B (h_{B})

__Solution :__

When arriving at point B, the kinetic energy of ball at point B = 2 times gravitational potential energy at point B.

EK = 2 EP

½ m v^{2} = 2 m g h_{B}

½ v^{2} = 2 g h_{B}

v^{2} = 2(2)(10) h_{B}

v^{2 }= 40 h_{B}

Velocity (v) of ball when arrive at point B after free fall from point A :

v^{2} = 2 g h = 2(10)(90–h_{B}) = 20(90–h_{B})

Substitute v^{2} at above equation with v^{2} at this equation.

v^{2} = 40 h_{B}

20(90–h_{B}) = 40 h_{B}

1800–20 h_{B} = 40 h_{B}

1800 = 40 h_{B} + 20 h_{B}

1800 = 60 h_{B}

h_{B }= 1800 / 60

h_{B} = 30 meters

5. A 1-kg ball is released and slides down from point A to point C, as shown in figure below. If the acceleration due to gravity = 10 m.s^{-2}, what is the kinetic energy of ball when arrive at point C.

__Known :__

Mass of ball (m) = 1 kg

Acceleration due to gravity (g) = 10 m/s^{2}

The change in height (h) = 0.75 m

The initial potential energy (PE_{o}) = m g h = (1)(10)(0.75) = 7.5 Joule

Initial kinetic energy (KE_{o}) = ½ m v_{o}^{2} = ½ m (0)^{2} = 0

The final gravitational potential energy (PE_{t}) = m g h = m g (0) = 0

__Wanted :__ Kinetic energy of ball at point C

__Solution :__

The initial mechanical energy (ME_{o}) = The final mechanical energy (ME_{t})

The initial gravitational potential energy (PE_{o}) + the initial kinetic energy (KE_{o}) = The final gravitational potential energy (PE_{t}) + the final kinetic energy (KE_{t})

7.5 Joule + 0 = 0 + KE_{t}

7.5 Joule = KE_{t }

Kinetic energy of ball at point C = 7.5 Joule.

6. A 2-kg ball is released from point A, as shown in figure below. The curve plane is smooth. If the acceleration due to gravity is 10 m.s^{-2}, what is the kinetic energy of ball at point B.

__Known :__

Mass of ball (m) = 2 kg

Acceleration due to gravity (g) = 10 m/s^{2 }

The change in height (h) = 120 cm = 1.2 meters

The initial gravitational potential energy (PE_{o}) = m g h = (2)(10)(1.2) = 24 Joule

The initial kinetic energy (KE_{o}) = ½ m v_{o}^{2} = ½ m (0)^{2} = 0

The final gravitational potential energy (PE_{t}) = m g h = m g (0) = 0

__Wanted :__ Kinetic energy of ball at point B

__Solution :__

The mechanical energy at point A (initial mechanical energy) = the mechanical energy at point B (the final mechanical energy)

The initial mechanical energy (ME_{o}) = the final mechanical energy (ME_{t})

The initial gravitational potential energy (PE_{o}) + the initial kinetic energy (KE_{o}) = the final gravitational potential energy (PE_{t}) + the final kinetic energy (KE_{t})

24 Joule + 0 = 0 + KE_{t}

24 Joule = KE_{t }

Kinetic energy (KE) of ball at point C = 24 Joule.

7. An object has a fixed mechanical energy, greater kinetic energy, and a smaller gravitational potential energy. The object is…

A. At rest

B. move upward

C. move downward

D. Accelerated upward

Solution :

Kinetic energy is getting bigger means the velocity of the object is getting bigger and the gravitational potential energy is smaller means that the height of the object from the soil surface is smaller. This happens when the object moves from a certain height down to the ground.

The relationship between speed and kinetic energy is shown by the kinetic energy formula:

KE = ½ m v^{2}

*Description of formula: KE = kinetic energy, m = mass, v = speed*

The relationship between height and gravitational potential energy is represented by the gravitational potential energy formula:

PE = m g h

*Description of formula: PE = gravitational potential energy, m = mass, g = acceleration of gravity, h = height*

The correct answer is C.

**What is mechanical energy?****Answer:**Mechanical energy is the sum of an object’s kinetic energy and potential energy. Kinetic energy relates to an object’s motion, while potential energy relates to its position or configuration.

**What does the principle of conservation of mechanical energy state?****Answer:**The principle states that in an isolated system with only conservative forces acting, the total mechanical energy remains constant. That means the sum of kinetic and potential energy does not change over time.

**How do conservative and non-conservative forces affect the conservation of mechanical energy?****Answer:**Conservative forces, like gravity, do not change the total mechanical energy of a system. They might convert potential energy to kinetic energy or vice-versa, but the sum remains constant. Non-conservative forces, like friction, can dissipate mechanical energy, typically transforming it into thermal energy.

**Why is mechanical energy not conserved in real-world scenarios involving friction?****Answer:**Friction is a non-conservative force. As objects move against each other, friction converts some of the system’s mechanical energy into thermal energy, leading to a decrease in the total mechanical energy of the system.

**How is the conservation of mechanical energy demonstrated in a swinging pendulum?****Answer:**In the absence of air resistance (a non-conservative force), as the pendulum swings upwards, its kinetic energy is converted into gravitational potential energy. At the highest point of its swing, it has maximum potential energy and minimal kinetic energy. As it swings back down, this potential energy is converted back into kinetic energy. The total mechanical energy remains constant, even though the forms of energy are interchanging.

**Why does a bouncing ball eventually come to rest even in the absence of external forces?****Answer:**When a ball bounces, not all of its kinetic energy is converted back into potential energy. Some energy is lost, often as sound or as deformation of the ball, and especially due to internal friction within the material of the ball. These are forms of non-conservative forces at play, which reduce the ball’s mechanical energy with each bounce until it comes to rest.

**In a roller coaster, where is the mechanical energy highest and lowest?****Answer:**The total mechanical energy of the roller coaster remains constant if we neglect air resistance and friction. However, at the highest point, the potential energy is at its maximum while the kinetic energy is at its minimum. As it descends, potential energy converts to kinetic energy, so at the lowest point, the kinetic energy is at its maximum and the potential energy is at its minimum.

**How does the conservation of mechanical energy help in understanding planetary orbits?****Answer:**Planets orbiting a star like the Sun have both kinetic energy (due to their motion) and gravitational potential energy (due to their position relative to the Sun). As a planet moves closer to the Sun, it speeds up, increasing its kinetic energy while its gravitational potential energy decreases, and vice versa. The total mechanical energy of the planet remains constant over time, assuming other forces (like from other planets) are negligible.

**Why does a satellite in a stable orbit not fall into the Earth?****Answer:**While the satellite is indeed being pulled toward the Earth due to gravity, its forward kinetic energy keeps it moving in its orbit. The balance of this kinetic energy and gravitational potential energy keeps the satellite in a stable orbit with constant mechanical energy.

**If a skier starts from rest at the top of a frictionless hill and then skis down, how will her speed at the bottom compare to if she started partway down the hill?**

**Answer:**On a frictionless hill, the skier’s total mechanical energy is conserved. If she starts from the very top, she’ll have more potential energy to begin with than if she started partway down. This potential energy will convert entirely into kinetic energy as she descends. Thus, she’ll be faster at the bottom if she starts at the top than if she started partway down.