A block of mass 3.5 kg slides down a frictionless inclined...

A block of mass 3.5 kg slides down a frictionless inclined plane of length 6.4 $\mathrm{m}$ that makes an angle of $30^{\circ}$ with the horizontal. If the block is released from rest at the top of the incline, what is its speed at the bottom? (A) 5.0 $\mathrm{m} / \mathrm{s}$ (B) 6.4 $\mathrm{m} / \mathrm{s}$ (C) 8.0 $\mathrm{m} / \mathrm{s}$ (D) 10 $\mathrm{m} / \mathrm{s}$

A block of mass 3.5 kg slides down a frictionless inclined plane of length 6.4 $\mathrm{m}$ that makes an angle of $30^{\circ}$ with the horizontal. If the
block is released from rest at the top of the incline, what is its speed at the bottom?
(A) 5.0 $\mathrm{m} / \mathrm{s}$
(B) 6.4 $\mathrm{m} / \mathrm{s}$
(C) 8.0 $\mathrm{m} / \mathrm{s}$
(D) 10 $\mathrm{m} / \mathrm{s}$

Key Concepts

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (like friction), the total mechanical energy of a system (the sum of kinetic and potential energy) remains constant. When an object moves from a higher elevation to a lower one, the gravitational potential energy lost is converted into kinetic energy, allowing one to compute the final speed of the object.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, and it is given by one half of the mass times the velocity squared. In energy conservation problems, the kinetic energy at the bottom is determined by the conversion of potential energy as the object descends.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. It is defined as the product of the mass, gravitational acceleration, and the height above a reference level. This energy is converted to kinetic energy during free fall or sliding down an incline.

Trigonometry in Physics

In problems involving inclined planes, trigonometry is used to relate the length of the incline to the vertical height. The sine of the angle of the incline gives the fraction of the incline’s length that corresponds to the vertical displacement, which is essential for calculating the change in gravitational potential energy.

Transcript

Please give Ace some feedback