Understand the setup:
The box sits on the trolley floor. When the trolley accelerates, the only horizontal force available to accelerate the box along with it is static friction from the trolley floor.
Forces on the box (horizontal):
Normal force N = mg = 15 × 10 = 150 N
Maximum static friction: \(f_{max} = \mu_s \times N = 0.12 \times 150 = 18 \text{ N}\)
Apply Newton's Second Law to the box:
For the box to stay stationary relative to trolley, friction must provide its acceleration:
$$f = ma \implies a_{max} = \frac{f_{max}}{m} = \frac{\mu_s mg}{m} = \mu_s g$$
Calculate:
$$a_{max} = \mu_s \times g = 0.12 \times 10 = \mathbf{1.2 \text{ m/s}^2}$$
Note: mass of box (15 kg) cancels out and is irrelevant! ✓
Friction is a contact force that opposes relative motion (or tendency of relative motion) between two surfaces in contact. It acts parallel to the surface of contact and arises due to microscopic interlocking of surface irregularities and intermolecular adhesive forces at contact points.
Friction is not always an undesirable force. In this problem, friction is the essential force that allows the box to accelerate with the trolley. Without friction, the box would remain stationary in the ground frame while the trolley slides under it. Walking, driving, and holding objects all depend on friction.
Static Friction (fₛ): Acts when surfaces are not sliding. It is a self-adjusting force — it takes whatever value is needed to prevent relative motion, up to a maximum of μₛN. It is always equal and opposite to the applied force (or tendency-causing force) until the maximum is reached.
Kinetic (Sliding) Friction (fₖ): Acts when surfaces are sliding. Its magnitude is μₖN — constant regardless of speed or area. Since μₖ < μₛ always, kinetic friction is less than maximum static friction. This is why it is easier to keep an object sliding than to start the slide.
Rolling Friction: Acts on rolling objects. Much smaller than sliding friction — this is why wheels are used in vehicles. Rolling friction coefficient is very small (~ 0.001–0.005 for rubber on road).
📌 Law 1: Friction force is proportional to normal reaction: f = μN
📌 Law 2: Friction is independent of area of contact
📌 Law 3: Kinetic friction is independent of speed of sliding
📌 Law 4: μₛ > μₖ always (static > kinetic coefficient)
The coefficient of friction μ is dimensionless and depends on the nature of the two surfaces in contact. It does not depend on mass, area, or speed. Typical values: rubber on concrete ≈ 0.7, steel on steel ≈ 0.4, ice on ice ≈ 0.03.
When an object rests on an accelerating surface, friction is the only horizontal force available to accelerate the object. The analysis is always from the ground (inertial) frame:
For object on accelerating surface:
\(f = ma\) (friction provides acceleration)
\(a_{max} = \mu_s g\) (when f = f_max)
Key insight: the maximum acceleration is μₛg, independent of mass. This is because both maximum friction force (μₛmg) and the required force (ma) are proportional to mass, so mass cancels.
In the reference frame of the accelerating trolley (non-inertial frame), a pseudo force acts on the box in the direction opposite to the trolley's acceleration. For the box to remain stationary in the trolley's frame, static friction must balance this pseudo force:
\(f_s \geq ma_{trolley}\)
\(\mu_s mg \geq ma_{trolley}\)
\(a_{trolley} \leq \mu_s g\)
Both the ground frame and trolley frame give the same condition: maximum acceleration = μₛg. Always choose the inertial (ground) frame for cleaner analysis in NEET.
The angle of friction (λ) is defined as tan λ = μ, where μ is the coefficient of friction. The resultant of normal force and friction force makes angle λ with the normal. The angle of repose (θ) is the maximum angle of inclination of a surface for which an object placed on it does not slide: tan θ = μₛ. Therefore angle of repose = angle of friction for static conditions.
📌 Object on trolley/conveyor belt: a_max = μg (this problem)
📌 Object on incline: N = mg cos θ, friction = μmg cos θ
📌 Two blocks, one on top of other: Friction between blocks limits relative motion
📌 Object being pushed horizontally: As normal force increases, friction increases
⚠️ Mass is irrelevant for maximum acceleration in this type of problem — always cancels out.
⚠️ Normal force = mg only when surface is horizontal and no vertical component of other forces acts.
⚠️ Static friction adjusts itself — it equals the required force up to μₛN, not always equal to μₛN.
⚠️ The 15 kg mass given in this question is a deliberate distractor — it does not affect the answer.