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Physics

Magnetic Suspension

A weight hangs near a magnet while a taut string runs down to a base. Explore how magnetic attraction, tension, and gravity can hold it at an angle.
Source code

Read the setup

In the supplied image, a small object sits below and to the right of an apparent magnet on a stand. A visible string connects the object to an anchor farther down and to the right. Assuming the object is stationary and magnetically attracted, three forces explain its suspension.

  • Gravity pulls vertically downward.
  • The taut string pulls toward the lower anchor: downward and rightward.
  • Magnetic attraction supplies the upward and leftward force needed to balance both.

“Invisible magnetic field lines” are a way to visualize a field, not physical threads holding the object. Here the real string constrains the motion and prevents the object from simply moving toward the magnet. The image alone does not reveal its mass, composition, or magnetic properties.

Find the required magnetic pull

Change the mass, string angle, and tension to calculate the magnetic force required for equilibrium.

Forces on the tethered weightMagnetic attraction points upward and left. Gravity points down. Tension points down and right along the string to the lower anchor. All three forces sum to zero.HorizontalLower anchorWeightMagnetic pull
Solid blue: magnetic attraction. Long dashes: tension. Dots: gravity. Arrow lengths share a scale within each diagram; the scale resets when inputs change. The faint line shows the string, not a field line.

Mass: 20 g20\,\mathrm{g}

String angle below horizontal: 60∘60^\circ

Tension relative to weight: T/(mg)=1.0T/(mg)=1.0

This calculates the pull needed for a chosen configuration. It does not simulate a particular magnet or predict a stable hanging position. The angle arc is measured from the horizontal to the string.

Weight
0.1962 N0.1962\,\mathrm{N}
String tension
0.1962 N0.1962\,\mathrm{N}
Required magnetic pull
0.3790 N0.3790\,\mathrm{N}

Resolve the three forces

Treat the object as a particle and the string as massless. Take right as positive xx and up as positive yy. The angle θ\theta is measured below the rightward horizontal, along the string toward the anchor. Let TT be the tension and mm the mass.

For static equilibrium, the vector sum of all forces is zero. By Newton’s second law, the object has zero acceleration:

∑F=ma=0{\textstyle\sum} \mathbf F=m\mathbf a=\mathbf 0Fmag+T+mg=0\mathbf F_{\mathrm{mag}}+\mathbf T+m\mathbf g=\mathbf 0Fmag,x=−Tcos⁡θ,Fmag,y=mg+Tsin⁡θF_{\mathrm{mag},x}=-T\cos\theta,\qquad F_{\mathrm{mag},y}=mg+T\sin\thetaFmag=(Tcos⁡θ)2+(mg+Tsin⁡θ)2F_{\mathrm{mag}}=\sqrt{(T\cos\theta)^2+(mg+T\sin\theta)^2}

The magnetic force must lift more than the weight alone because the string also pulls downward. The string angle alone cannot determine tension or magnetic force; another measurement or a magnetic-field model is needed.

At the lab defaults, m=20 gm=20\,\mathrm{g}, T=mg=0.1962 NT=mg=0.1962\,\mathrm{N}, and θ=60∘\theta=60^\circ. The required horizontal component is −0.0981 N-0.0981\,\mathrm{N}, the upward component is 0.3661 N0.3661\,\mathrm{N}, and the total magnetic pull is approximately 0.3790 N0.3790\,\mathrm{N}. These are illustrative values, not measurements from the image.

What the magnetic field actually does

A magnet can attract a ferromagnetic object by magnetizing it. A small permanent magnet also experiences a force in a nonuniform external field. In the fixed magnetic dipole approximation, the potential energy and force are

U=−μ⋅B,F=∇(μ⋅B)U=-\boldsymbol\mu\cdot\mathbf B,\qquad \mathbf F=\nabla(\boldsymbol\mu\cdot\mathbf B)

Here μ\boldsymbol\mu is the magnetic dipole moment and B\mathbf B is the external magnetic field. Spatial variation matters: a uniform field can exert a turning effect on a fixed dipole but produces no net translational force. A force arrow therefore need not follow a field line.

An extended steel object can have induced, nonuniform magnetization; predicting its pull needs its geometry and material response as well as the nearby field. The particle calculation above only gives the force that would be required, not the force a particular magnet can supply.

Feynman Lectures: forces on a current loop and energy of a dipole

Balance is only the first condition

Zero net force means no acceleration at that instant. Stable suspension also needs a restoring response to small displacements allowed by the string. That depends on how the magnetic force changes with position. The extended object must also have zero net torque; its tilt is not determined by the particle model.

  • Remove the magnet: gravity and the lower string cannot maintain the pictured position.
  • Cut a taut string: at the initial instant, the remaining force points opposite to the former tension, upward and leftward.
  • Move the magnet: the object may shift, lose tension, or reach the magnet, depending on the new field and geometry.

This is tethered magnetic suspension. Demonstrating a stable position requires more information than a still image or a force balance alone.