A person is pulling a block of mass m with a force equal to...

A person is pulling a block of mass $m$ with a force equal to its weight directed $30^{\circ}$ above the horizontal plane across a rough surface, generating a friction $f$ on the block. If the person is now pushing downward on the block with the same force $30^{\circ}$ above the horizontal plane across the same rough surface, what is the friction on the block? $\left(\mu_{k} \text { is the coefficient of kinetic friction across the surface.) }\right.$ (A) $f$ (B) 1.5$f$ (C) 2$f$ (D) 3$f$

A person is pulling a block of mass $m$ with a force equal to its weight
directed $30^{\circ}$ above the horizontal plane across a rough surface,
generating a friction $f$ on the block. If the person is now pushing
downward on the block with the same force $30^{\circ}$ above the horizontal
plane across the same rough surface, what is the friction on the block?
$\left(\mu_{k} \text { is the coefficient of kinetic friction across the surface.) }\right.$
(A) $f$
(B) 1.5$f$
(C) 2$f$
(D) 3$f$

Key Concepts

Effect of Force Angle

The angle at which a force is applied significantly influences its impact on the system. The horizontal component drives or opposes motion, while the vertical component adjusts the normal force. When the applied force has an upward vertical component, it reduces the normal force, thereby lowering friction, and when it has a downward component, it increases the normal force, thereby increasing friction.

Force Decomposition

Force decomposition involves breaking a force vector into its horizontal and vertical components using trigonometric functions such as sine and cosine. This process is essential for analyzing how a single force affects motion in different directions, particularly in situations where forces contribute variably to movement and changes in normal force.

Kinetic Friction

Kinetic friction is the resistive force acting opposite to the direction of motion when two surfaces slide against one another. It is calculated as the product of the coefficient of kinetic friction and the normal force, so any change in the normal force directly affects the magnitude of kinetic friction.

Normal Force

Normal force is the component of contact force perpendicular to the surface. It is crucial in friction problems because the frictional force depends directly on the normal force. When additional forces are applied with vertical components, they either decrease or increase the normal force depending on their direction relative to the weight of the object.

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