Explore conservative and non-conservative forces, energy transfer, Hooke's law and the potential energy stored in a stretched or compressed spring.
A force is conservative if the work done by it between two points depends only on the initial and final positions, not on the path followed. Work done by a conservative force over a closed path is zero.
For a conservative force, W = −ΔU = Uinitial − Ufinal, where U is potential energy.
Examples include gravitational force and the ideal spring force.
A force is non-conservative when its work depends on the path. Friction and air resistance are common examples. They transfer some mechanical energy into thermal energy and other forms that are not easily recovered as mechanical work.
Answer: A conservative force does work that depends only on the starting and ending positions; its work over a closed path is zero. Gravity is an example.
Answer: The work done by gravity depends only on the change in height, not on the route taken between the two points.
Answer: Work done by friction depends on the path length and surface conditions. It converts some mechanical energy into heat, so its work over a closed path is not generally zero.
Answer: Mechanical energy may be transformed into other forms, such as thermal energy or sound. Total energy is conserved, but mechanical energy alone may decrease.
Answer: Moving objects would not slow down because of friction. They could continue moving at constant velocity unless another force acted. Walking, gripping objects and braking vehicles would also be much more difficult without friction.
When a spring is stretched or compressed within its elastic limit, it exerts a restoring force. For an ideal spring, the magnitude of this force is proportional to the extension or compression.
Hooke's law: F = kx
Here, k is the spring constant (N/m) and x is the extension or compression from the natural length (m). The restoring force acts opposite to the displacement, so in signed form F = −kx.
The energy stored in a spring stretched or compressed by x is:
U = ½kx²
This is the work required to stretch the spring slowly from x = 0 to x. It is stored as elastic potential energy, provided the elastic limit is not exceeded.
U = ½kx² = ½ × 100 × (0.2)²
= 50 × 0.04 = 2 J.
U = ½ × 200 × (0.1)² = 100 × 0.01 = 1 J.
F = kx, so x = F/k = 30/150 = 0.2 m.
U = ½ × 400 × (0.05)² = 200 × 0.0025 = 0.5 J.
Answer: For a conservative force, work between two points is path-independent and work over a closed path is zero. For a non-conservative force, work depends on the path; friction is a common example.
Answer: The spring's elastic potential energy is transferred mainly into kinetic energy as it moves. Depending on the situation, some energy may also become sound and thermal energy.
Since U = ½kx², if x becomes 2x, then U′ = ½k(2x)² = 4(½kx²).
Answer: The stored energy becomes four times the original value, provided the spring remains within its elastic limit.
F = kx = 80 × 0.15 = 12 N. The restoring force acts opposite to the displacement.
Answer: Friction opposes the book's motion and transfers its kinetic energy into thermal energy in the book and table. The book slows down and eventually stops.
U = ½kx² = ½ × 250 × (0.08)² = 125 × 0.0064 = 0.8 J.
Hang a spring vertically, record its natural length, and add known loads in small steps. Measure the extension after each addition. Record force (weight) and extension in a table. Within the elastic limit, a graph of applied force against extension should be approximately a straight line; its gradient gives the spring constant k.
Do not exceed the spring's elastic limit. Repeat measurements and use consistent units: force in newtons and extension in metres.