This question relates to the Work-Energy Theorem, which states that the net work done on an object is equal to the change in its kinetic energy. We need to determine the sign of the work done when both work is performed and kinetic energy increases.
where \(W_{\text{net}}\) is the net work done on the object, and \(\Delta K\) is the change in its kinetic energy.
Kinetic energy \(K\) is always a non-negative quantity, calculated as \(K = \frac{1}{2}mv^2\), where \(m\) is mass and \(v\) is speed.
A) Positive — As per the Work-Energy Theorem, if the kinetic energy of an object increases, the net work done on it must be positive.
The question asks about the relationship between force and displacement when no work is done. We need to recall the definition of work done by a force and analyze the conditions under which it becomes zero.
D) The displacement is perpendicular to the force. This is the condition for zero work when both force and displacement are non-zero, as derived from the work formula \(W = F d \cos \theta\). If \(\theta = 90^\circ\), then \(\cos 90^\circ = 0\), making \(W = 0\).
To determine where work is being done, we need to recall the definition of work in physics. Work is done when a force causes a displacement in the direction of the force. Mathematically, work \(W\) is defined as the dot product of the force vector \(\vec{F}\) and the displacement vector \(\vec{d}\), or \(W = Fd \cos\theta\), where \(\theta\) is the angle between the force and displacement.
Work done while lifting the box: When a person lifts a box from the ground, they apply an upward force against gravity. The displacement of the box is also in the upward direction. Since the force and displacement are in the same direction (or nearly so), the angle \(\theta = 0^\circ\), and \(\cos 0^\circ = 1\). Therefore, work is done against gravity.
\[ W_{\text{lifting}} = F_{\text{up}} \times d_{\text{up}} \]Work done while carrying the box horizontally: Once the box is lifted, the person carries it horizontally. To hold the box, the person applies an upward force (against gravity). However, the displacement is horizontal. The angle between the upward force and the horizontal displacement is \(90^\circ\). Since \(\cos 90^\circ = 0\), the work done by the person's upward force in carrying the box horizontally is zero.
\[ W_{\text{carrying}} = F_{\text{up}} \times d_{\text{horizontal}} \times \cos 90^\circ = 0 \]It's important to note that while the person expends energy to carry the box (due to muscle tension and metabolic processes), from a physics perspective, no work is done on the box itself by the force holding it against gravity during horizontal motion.
Correct Option: C) While lifting the box
The question asks for the SI unit of work. To answer this, we need to recall the definition of work in physics and its standard unit in the International System of Units (SI). We will then evaluate each option to identify the correct unit for work and explain why the other options are incorrect.
Correct Option: B) Joule is the SI unit of work. One Joule is defined as the work done when a force of one Newton displaces an object by one meter in the direction of the force. Mathematically, \(1 \text{ J} = 1 \text{ N} \cdot \text{m}\).
The question asks about the conditions under which work is done when a force is applied to an object. Work in physics is defined as the product of the force applied and the displacement of the object in the direction of the force. We will use the formula for work done to evaluate each option.
A) The object moves in the direction of the force
To determine the work done, we need to recall the definition and formula for work in physics. Work is done when a force causes a displacement of an object. The key here is the presence or absence of displacement.
Definition of Work: In physics, work done by a constant force is defined as the product of the magnitude of the force, the magnitude of the displacement, and the cosine of the angle between the force and the displacement vectors.
Formula for Work Done: The mathematical formula for work done (\(W\)) is given by:
\[ W = F \cdot d \cdot \cos(\theta) \]
Where:
Applying the Condition: The question states that "no displacement occurs." This means that the magnitude of the displacement, \(d\), is zero.
\[ d = 0 \]
Calculating Work Done: Substitute \(d = 0\) into the work formula:
\[ W = F \cdot (0) \cdot \cos(\theta) \]
\[ W = 0 \]
Conclusion: If the displacement is zero, the work done is always zero, regardless of the force applied or the angle between the force and the (non-existent) displacement.
C) Zero work — As derived from the formula \(W = Fd\cos\theta\), if the displacement \(d\) is zero, then the work done \(W\) must be zero.
A) Positive work — Positive work is done when the force has a component in the direction of displacement, i.e., when \(0 \le \theta < 90^\circ\). This requires a non-zero displacement.
B) Negative work — Negative work is done when the force has a component opposite to the direction of displacement, i.e., when \(90^\circ < \theta \le 180^\circ\). This also requires a non-zero displacement.
D) Insufficient information to determine — The information provided (no displacement) is sufficient to determine the work done, as it directly leads to zero work based on the definition.