The spring constant, often denoted by \(k\), is a measure of the stiffness of a spring. It relates the force applied to a spring to the displacement it causes. We can determine its SI unit by using Hooke's Law.
Recall Hooke's Law: Hooke's Law states that the force \(F\) required to extend or compress a spring by some distance \(x\) is proportional to that distance. Mathematically, it is expressed as:
\[ F = -kx \]where \(F\) is the restoring force exerted by the spring, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position. The negative sign indicates that the restoring force is in the opposite direction to the displacement. For calculating the magnitude of the spring constant, we can consider the magnitude of the force:
\[ |F| = k|x| \]Rearrange the formula to solve for \(k\):
\[ k = \frac{|F|}{|x|} \]Substitute the SI units for force and displacement:
Calculate the SI unit for \(k\):
\[ \text{Unit of } k = \frac{\text{Unit of } F}{\text{Unit of } x} = \frac{\text{Newton}}{\text{meter}} = \text{N/m} \]This can also be written as \(\text{N} \cdot \text{m}^{-1}\) or \(\text{Nm}^{-1}\).
Correct Option: B) Nm \(^{-1}\) is the correct SI unit for the spring constant, derived from Hooke's Law as Force divided by displacement (Newton per meter).
The question asks to identify the type of collision where both linear momentum and kinetic energy are conserved. We need to recall the definitions of different types of collisions based on the conservation laws applied to them.
C) Elastic collision is the type of collision where both the linear momentum and the kinetic energy of the system remain conserved. This is the defining characteristic of an elastic collision.
The question asks to identify the type of energy associated with an object's motion. We need to recall the definitions of different forms of energy.
Correct Option: B) Kinetic energy — This is the energy an object possesses due to its motion. For example, a moving car, a flying bird, or a rolling ball all have kinetic energy.
To identify vector quantities, we need to understand the definition of a vector. A vector quantity is a physical quantity that has both magnitude and direction. In contrast, a scalar quantity has only magnitude. We will examine each option to determine if it possesses both magnitude and direction.
D) All the above. Displacement, velocity, and acceleration are all physical quantities that possess both magnitude and direction, making them vector quantities.
The problem asks for the work done by the table surface on a ball moving without slipping on a frictionless inclined table. We need to analyze the forces acting on the ball and the definition of work done.
Identify the forces: When a ball is on an inclined table, the forces acting on it are gravity (downwards), and the normal force from the table (perpendicular to the surface).
Consider the normal force: The table surface exerts a normal force on the ball. This force is always perpendicular to the surface of contact. Since the ball is moving on the surface, its displacement is always along the surface.
Recall the definition of work done: Work done by a force is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{d} = |\vec{F}| |\vec{d}| \cos\theta \]where \(\theta\) is the angle between the force vector and the displacement vector.
Apply to the normal force: For the normal force, the force vector is perpendicular to the surface. The displacement vector of the ball is along the surface. Therefore, the angle \(\theta\) between the normal force and the displacement is \(90^\circ\).
Calculate the work done: Substituting \(\theta = 90^\circ\) into the work done formula:
\[ W_{\text{normal}} = F_{\text{normal}} \cdot d \cdot \cos(90^\circ) \]Since \(\cos(90^\circ) = 0\), the work done by the normal force is zero.
Consider the friction: The problem states the table is frictionless and the ball moves without slipping. If the table is frictionless, there is no frictional force acting on the ball from the table surface. Therefore, the work done by friction is also zero.
Total work done by the table surface: The table surface exerts only the normal force (and no friction). Since the work done by the normal force is zero, the total work done by the table surface on the ball is zero.
Correct Option: C) Zero
The concept of work in physics is defined as the transfer of energy to or from an object by applying a force that causes displacement. It is closely related to changes in the object's energy state.
where \(\theta\) is the angle between the force vector and the displacement vector.
More broadly, work done can lead to changes in other forms of energy as well, such as potential energy, or an overall increase in the total mechanical energy of the system due to external mechanical influence.
B) When there is an increase in energy because of mechanical influence. This option correctly captures the essence of work. Work is a mechanism for transferring energy. If work is done on the body, its energy increases. This increase in energy is a direct result of a mechanical influence (force causing displacement).