Properties of a Couple

A couple is formed by two parallel forces that are equal in magnitude and opposite in direction, separated by a perpendicular distance. A couple produces a rotational effect without causing any translational motion.

Analyzing Statements about a Couple

Let's examine each statement to determine which one is incorrect regarding the properties of a couple.

• Statement 1: A couple cannot be balanced by a single force.

  A single force applied to a body generally causes both translation (movement in a straight line) and rotation. A couple, however, only causes rotation. To balance the rotational effect of a couple, you need another couple producing an equal and opposite moment. A single force cannot counteract only the rotation without also causing translation. Therefore, this statement is correct.

• Statement 2: The couple can be replaced by another pair of forces whose rotational effects are the same.

  The rotational effect of a couple is measured by its moment. The moment of a couple is the product of the magnitude of one of the forces and the perpendicular distance between the forces (moment arm). If you have another pair of forces with a different magnitude and different separation, but their product (Force $\times$ distance) is the same, they will produce the same moment and thus the same rotational effect. Therefore, a couple can be replaced by an equivalent couple. This statement is correct.

• Statement 3: The translator effect of a couple on the body is zero.

  A couple consists of two forces, $\vec{F}$ and $-\vec{F}$, acting at different points. The net force on the body is the vector sum of these forces, which is $\vec{F} + (-\vec{F}) = \vec{0}$. Since the net force is zero, according to Newton's second law ($\vec{F}_{net} = m\vec{a}$), there is no net acceleration in any direction, meaning there is no translational effect. The body will not move in a straight line unless it is already moving due to other forces. Therefore, this statement is correct.

• Statement 4: The moment of a couple about different points in its plane is different.

  One of the unique properties of a couple is that its moment is constant about any point in the plane containing the couple. Let the two forces be $F$ and $-F$ separated by a distance $d$. If you take moments about a point P in the plane, the total moment is the sum of the moments of the two forces about P. Regardless of the position of P, the sum of the moments will always be $F \times d$. The moment of a couple is a free vector, meaning its effect is the same no matter where it is applied in the plane. Therefore, the statement that the moment is different about different points is incorrect.

Based on the analysis, the incorrect statement is that the moment of a couple about different points in its plane is different.