To determine when the torque produced by a force is maximum, we need to recall the formula for torque and analyze its dependence on the angle between the force vector and the position vector (line joining the point to the force).
Definition of Torque: Torque (\(\tau\)) is the rotational equivalent of force. It is defined as the cross product of the position vector (\(\vec{r}\)) from the pivot point to the point where the force is applied, and the force vector (\(\vec{F}\)).
Mathematical Formula for Torque: The magnitude of torque is given by:
\[ \tau = r F \sin(\theta) \]where:
Analyzing for Maximum Torque: For a given force \(F\) and distance \(r\), the torque \(\tau\) will be maximum when the \(\sin(\theta)\) term is maximum.
Maximum Value of Sine Function: The maximum value of \(\sin(\theta)\) is 1. This occurs when \(\theta = 90^\circ\) (or \(\pi/2\) radians).
Conclusion: Therefore, the torque produced by a force about a point is maximum when the angle between the force and the line joining the point to the force is \(90^\circ\).
Correct Option: C) 90 degrees
When the angle \(\theta\) between the position vector \(\vec{r}\) and the force vector \(\vec{F}\) is \(90^\circ\), \(\sin(90^\circ) = 1\). This makes the torque \(\tau = rF(1) = rF\), which is the maximum possible value for a given \(r\) and \(F\).
The question asks to identify the type of equilibrium represented by the point of suspension of a pendulum. To answer this, we need to understand the definitions of different types of equilibrium: stable, unstable, and neutral.
C) Neutral equilibrium. The point of suspension of a pendulum is a fixed point. If we consider its 'equilibrium' in a conceptual sense, any slight displacement of this point (e.g., if it were allowed to slide along a horizontal track) would result in it remaining in that new position without tending to return or move further away. Its potential energy would remain constant with such a displacement. Therefore, it exhibits neutral equilibrium.
The question asks about the location of the center of gravity of a body. The center of gravity is a theoretical point where the entire weight of an object appears to act. For a uniformly dense and symmetrically shaped body, its center of gravity coincides with its geometrical center.
A) Always coincides with its geometrical center
This statement is generally true for homogeneous (uniformly dense) and symmetrically shaped bodies. In physics problems, unless stated otherwise, bodies are often assumed to be homogeneous and symmetrical for simplicity. For such bodies, the center of gravity is indeed at the geometrical center.
The center of gravity can lie outside the physical boundaries of the body, but it doesn't always lie above the surface. For example, in a hollow ring, the center of gravity is in the empty space at the center of the ring, which is not "above the surface" in a general sense. For a stable object like a cube, the CG is well within its volume, not above its surface.
Similar to option B, the center of gravity can be within the body, but it doesn't always lie below the surface. For many objects, it's within the body itself. For example, the CG of a sphere is at its center, which is inside the sphere, not specifically "below the surface."
While the density of the material affects the distribution of mass, and thus the location of the center of mass (which is often coincident with the center of gravity), the statement "depends on the material" is too vague and not the primary defining characteristic of the CG's location relative to the body's geometry. For a homogeneous body, the material's specific type (e.g., iron vs. wood) doesn't change the fact that its CG is at its geometrical center, assuming the shape and uniform density are maintained.
This question tests the understanding of different types of equilibrium in physics. Stable equilibrium is characterized by a system's tendency to restore its original state after a small disturbance.
Correct Option: B) Return to its initial position when slightly disturbed
To determine the condition for rotational equilibrium, we need to understand what rotational equilibrium means. A body is in equilibrium when its state of motion (linear and rotational) is not changing. Rotational equilibrium specifically refers to the absence of change in rotational motion.
A) Net torque acting on the body must be zero
This is the precise definition of rotational equilibrium. If the net torque is zero, the body will either remain at rest rotationally or continue to rotate at a constant angular velocity.
This question asks about the definition of unstable equilibrium in physics. Equilibrium refers to a state where the net force and net torque on an object are zero. There are three types of equilibrium: stable, unstable, and neutral. These are distinguished by how a system responds to a small disturbance, specifically in relation to its potential energy.
A) Any disturbance shifts the body to a higher position of potential energy. This statement correctly describes unstable equilibrium. In an unstable equilibrium, the potential energy is at a local maximum. Any slight displacement from this point causes the body to move towards a state of lower potential energy, thus moving further away from the original equilibrium position.