Question

b4 - lmct18: block pushed on incline—work done
a block is pushed so that it moves up a ramp at constant speed. identify from
choices (i)-(iv) below the appropriate description for the work done by the specified
force while the block moves from point a to point b.
(i) is zero. (ii) is less than zero. (iii) is greater than zero.
(iv) could be positive or negative depending on the choice of coordinate systems.
(v) cannot be determined.
(a) the work done on the block by the hand. ____ explain your reasoning.
(b) the work done on the block by the normal force from the ramp. ____ explain your reasoning.
(c) the work done on the block by friction. ____ explain your reasoning.
(d) the work done on the block by the gravitational force. ____ explain your reasoning.
(e) the net work done on the block. ____ explain your reasoning.

Explanation:

• (a) Work done by the hand:

The force applied by the hand is in the direction of the block's displacement (from \(A\) to \(B\)). Using the work - formula \(W = F\cdot d\cdot\cos\theta\) (where \(\theta\) is the angle between the force \(F\) and displacement \(d\)), and since \(\theta = 0^{\circ}\) (force and displacement are in the same direction), \(\cos\theta=1\). So, \(W>0\).

• (b) Work done by the normal force:

The normal force \(N\) is perpendicular to the surface of the ramp. The displacement of the block is along the ramp. So, the angle \(\theta = 90^{\circ}\) between the normal force and displacement. Using \(W = F\cdot d\cdot\cos\theta\) and \(\cos(90^{\circ}) = 0\), so \(W = 0\).

• (c) Work done by friction:

Friction acts opposite to the direction of the block's motion. Let the force of friction be \(f\) and displacement be \(d\). The angle \(\theta=180^{\circ}\) between the frictional force and displacement. Using \(W = F\cdot d\cdot\cos\theta\) and \(\cos(180^{\circ})=- 1\), so \(W<0\).

• (d) Work done by gravitational force:

The gravitational force \(mg\) has a component \(mg\sin\alpha\) (where \(\alpha\) is the angle of the ramp with the horizontal) opposite to the direction of displacement. Let the displacement be \(d\). The angle \(\theta = 180^{\circ}\) between the component of gravitational force along the ramp and displacement. Using \(W = F\cdot d\cdot\cos\theta\) and \(\cos(180^{\circ})=-1\), so \(W<0\).

• (e) Net work done on the block:

Since the block moves at a constant speed, its kinetic energy change \(\Delta K = 0\). According to the work - energy theorem \(W_{net}=\Delta K\). So, \(W_{net}=0\).

Answer:

(a) (iii) is greater than zero.
(b) (i) is zero.
(c) (ii) is less than zero.
(d) (ii) is less than zero.
(e) (i) is zero.