Newton's Cradle

Arguably the best desk ornament. Here's how it works! 

Watch The Video:

Teachable Topics:

• Conservation of Momentum
• Conservation of Energy
• Elastic Collision

Theory:

In an elastic collision, both momentum (p) and kinetic energy (T) are conserved. That is

p_f = p_i and T_f = T_i

Newton's pendulum is five metal spheres suspended so they all touch one another in line. One or more spheres can be drawn back and released. The released spheres swing down and collide with the remaining spheres.

Each sphere has mass m, and the spheres collide essentially elastically. Of course, the fact that the spheres click when they connect means that some kinetic energy is being converted into sound energy and lost, but these losses are negligible in a single collision.

When one or more spheres are drawn back and released, it/they have mass M_i = m, 2m, 3m, 4m,depending on if one, two, three, or four spheres are drawn back.

At the instant just before these spheres collide with the other hanging spheres, they have a velocity (in the horizontal direction) of v_i. The momentum and kinetic energy are thus p_i = M_iv_i , and T_i =(1/2)M_iv_i².

An instant after the spheres collide, some mass of spheres (M_f) must keep moving with velocity (in the horizontal direction) v_f. Applying the conservation of momentum to this collision gives the following equation:

p_f = p_i

M_fv_f = M_iv_i

Conserving kinetic energy gives....

T_f = T_i

(1/2)M_fv_f² = (1/2)M_iv_i²

M_f = M_i(v_i/v_f)²

The red and green equations above can be combined to eliminate M_f.

M_i(v_i/v_f)² = M_i(v_i/v_f)

So the velocity of any moving spheres after the collision is the same as the velocity of the moving spheres before the collision. Substituting this into the red equation gives...

M_f = M_i(v_i/v_f)

M_f = M_i (v_i/v_i)

M_f = M_i

The number of spheres moving after the collision is the same as the number moving before the collsion

Apparatus:

• A Newton's Pendulum

Procedure:

• Pull back one of the spheres in the pendulum, then release.
• You can do this with two, three or four spheres.
• The demonstration works most smoothly if the sphere(s) is/are released from a small angle.