The statue of Isaac Newton is crated and moved from the Oxford Museum of Natural History and Science for cleaning. The mass of the statue and the crate is 150kg. As the statue slides down a ramp inclined at 40 degrees, the curator pushes up, parallel to the ramp’s surface, so that the crate does not accelerate. If the statue slides 3m down the ramp, and the coefficient of kinetic friction between the crate and the ramp is 0.54, calculate the work done on the crate by each of the following:
(a) the gravitational force,
(b) the curator,
(c) the friction force, and
(d) the noral force between the ramp and the crate.
The correct answer and explanation is :
We are given:
- Mass $m = 150 \, \text{kg}$
- Distance $d = 3.0 \, \text{m}$
- Incline angle $\theta = 40^\circ$
- Coefficient of kinetic friction $\mu_k = 0.54$
- Acceleration = 0 (crate moves at constant speed)
We will calculate the work done by:
(a) Gravitational Force
Gravitational force component parallel to the incline:
$$
F_g = mg \sin\theta = 150 \cdot 9.8 \cdot \sin(40^\circ) \approx 150 \cdot 9.8 \cdot 0.6428 \approx 944.5 \, \text{N}
$$
$$
W_g = F_g \cdot d = 944.5 \cdot 3 = \boxed{2833.5 \, \text{J}}
$$
(b) Curator’s Force
Since the crate moves at constant speed, total force along the incline = 0. So curator must balance both friction and gravity.
Friction force:
$$
f_k = \mu_k \cdot N = \mu_k \cdot mg \cos\theta = 0.54 \cdot 150 \cdot 9.8 \cdot \cos(40^\circ) \approx 0.54 \cdot 150 \cdot 9.8 \cdot 0.766 \approx 606.2 \, \text{N}
$$
Total force down incline:
$$
F_g + f_k = 944.5 + 606.2 = 1550.7 \, \text{N}
$$
The curator applies equal and opposite force:
$$
F_{\text{curator}} = 1550.7 \, \text{N} \quad \Rightarrow \quad W_c = -1550.7 \cdot 3 = \boxed{-4652.1 \, \text{J}}
$$
(c) Friction Force
Acts opposite to motion:
$$
W_f = -f_k \cdot d = -606.2 \cdot 3 = \boxed{-1818.6 \, \text{J}}
$$
(d) Normal Force
Always perpendicular to motion (incline surface). So:
$$
W_N = 0 \quad \boxed{0 \, \text{J}}
$$
🧠 Explanation (300 words):
This problem involves computing the work done by various forces as a crate containing a statue slides down a ramp at a constant speed. Since there is no acceleration, the net force along the ramp is zero — meaning the curator must exert a force that exactly balances the sum of gravitational and frictional forces acting down the incline.
First, we calculate the work done by gravity, which is the component of the object’s weight parallel to the incline. As the crate moves in the direction of this force, the work is positive.
Next, the curator’s force is applied in the opposite direction of motion (up the incline), to prevent acceleration. This means the work done by the curator is negative, as it acts against the direction of displacement.
Friction also opposes the motion, and therefore does negative work. It depends on the normal force (the perpendicular force from the ramp surface), which we compute using the cosine of the incline angle.
Lastly, the normal force acts perpendicular to the direction of motion and does no work, since work involves displacement in the direction of the force.
Summing the work from all forces gives zero, consistent with the principle of energy conservation for an object moving at constant speed:
$$
W_g + W_c + W_f + W_N = 2833.5 – 4652.1 – 1818.6 + 0 \approx 0 \, \text{J}
$$