A block of mass 2 ·0 kg kept at rest on an inclined plane...

A block of mass $2 \cdot 0 \mathrm{~kg}$ kept at rest on an inclined plane of inclination $37^{\circ}$ is pulled up the plane by applying a constant force of $20 \mathrm{~N}$ parallel to the incline. The force acts for one second. (a) Show that the work done by the applied force does not exceed $40 \mathrm{~J} .$ (b) Find the work done by the force of gravity in that one second if the work done by the applied force is $40 \mathrm{~J} .(\mathrm{c})$ Find the kinetic energy of the block at the instant the force ceases to act. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$.

Key Concepts

Force Decomposition on an Inclined Plane

Force decomposition on an inclined plane is the process of breaking down a force, such as gravity, into components parallel and perpendicular to the plane. This concept is crucial for analyzing the influence of different forces on the motion of an object on a slope, as only the component parallel to the plane contributes directly to the work done in moving the object along the incline.

Motion on an Inclined Plane

Motion on an inclined plane involves analyzing forces acting on an object when it is located on a sloped surface. Key considerations include the component of gravitational force acting parallel to the incline and perpendicular to the surface, which affects both the work done by gravity and the acceleration of the object. This setup is fundamental in understanding how forces are decomposed in non-horizontal systems.

Kinematics under Constant Acceleration

Kinematics under constant acceleration deals with the equations that relate displacement, velocity, acceleration, and time when an object is moving with an unchanging acceleration. These relations are critical for determining how far an object travels under a specific force acting for a given duration, which in turn is used to compute the work done by that force.

Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. This principle provides a direct connection between the forces acting on an object and the resulting changes in its motion, making it an essential tool for solving problems where an object’s speed or energy state is altered by applied forces.

Work Done by a Force

Work is defined as the product of a constant force and the displacement of the point of application of the force in the direction of the force. It represents the energy transferred to or from an object via the application of force along the direction of its motion. This concept is central in analyzing scenarios where energy is imparted to or removed from a system by forces acting over a distance.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion and is calculated as one-half the mass of the object multiplied by the square of its velocity. It quantifies the energy that must be added to or removed from an object to change its speed, and its relationship with work underpins many energy conservation and motion problems.

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