# 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/s2. Known :

Mass (m) = 1 kg

The change in height (h) = 5 m

Acceleration due to gravity (g) = 10 m/s2

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

Solution :

(a) Kinetic energy

The initial mechanical energy = gravitational potential energy

MEo = PE = m g h = (1)(10)(5) = 50 Joule

The final mechanical energy = kinetic energy

MEt = KE = ½ m vt2

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

MEo = MEt

PE = KE

50 = KE

Kinetic energy (KE) = 50 Joule.

(b) Block’s velocity

The principle of conservation of mechanical energy :

The initial mechanical energy (MEo) = the final mechanical energy (MEt)

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

50 = ½ m v2

2(50) / m = v2

100 / 1 = v2

100 = v2

v = √100

v = 10 m/s

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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/s2 Known :

Mass (m) = 2 kg

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

Acceleration due to gravity (g) = 10 m/s2

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

MEo = PE = m g h = (2)(10)(8) = 160 Joule

The final mechanical energy = kinetic energy

MEt = KE = ½ m vt2

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

MEo = MEt

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 (MEo) = the final mechanical energy (EMt)

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

160 = ½ m v2

160 = ½ (2) v2

160 = v2

v = √160 = √(16)(10) = 4√10 m/s

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