# Application of conservation of mechanical energy for motion on curve surface – problems and solutions

1. A 1-kg block slides down on the smooth curved surface. Determine the kinetic energy and the velocity of the block at the lowest surface. Acceleration due to gravity is 10 m/s^{2}.

__Known :__

The change in height (h) = 5 m

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

Wanted: Kinetic energy (KE) and the velocity of the block.

__Solution :__

**(a) Kinetic energy**

__The initial mechanical energy = gravitational potential energy__

ME_{o} = PE = m g h = (1)(10)(5) = 50 Joule

__The final mechanical energy = kinetic energy__

ME_{t} = KE = ½ m v_{t}^{2}

Principle of conservation of mechanical energy states that the initial mechanical energy = the final mechanical energy :

ME_{o} = ME_{t}

PE = KE

50 = KE

Kinetic energy (KE) = 50 Joule.

**(b) Block’s velocity**

The principle of conservation of mechanical energy :

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

The gravitational potential energy (PE) = kinetic energy (KE)

50 = ½ m v^{2}

2(50) / m = v^{2}

100 / 1 = v^{2}

100 = v^{2}

v = √100

v = 10 m/s

2. A 2-kg object slides down without friction. What is the kinetic energy and the velocity of the object at 2 meters above the ground. Acceleration due to gravity is 10 m/s^{2}

__Known :__

Mass (m) = 2 kg

The change in height (h) = 10 – 2 = 8 m

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

__Wanted __: kinetic energy (KE) and velocity (v) at 2 meters above the ground.

__Solution :__

**(a) Kinetic energy at 2 meters above the ground**

__The initial mechanical energy = the gravitational potential energy__

ME_{o} = PE = m g h = (2)(10)(8) = 160 Joule

__The final mechanical energy = kinetic energy__

ME_{t} = KE = ½ m v_{t}^{2}

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

ME_{o} = ME_{t}

PE = KE

160 = KE

Kinetic energy (KE) at 2 meters above the ground is 160 Joule.

**(b) Object’s velocity at the lowest surface**

Principle of conservation of mechanical energy :

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

The gravitational potential energy (PE) = kinetic energy (KE)

160 = ½ m v^{2}

160 = ½ (2) v^{2}

160 = v^{2}

v = √160 = √(16)(10) = 4√10 m/s

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