Two blocks on meeting inclines are connected by a light string over an ideal pulley placed at the top where the slopes meet.
The string forces both blocks to have the same acceleration magnitude. In this calculator, the positive direction is defined
so that \(m_2\) moves down the right incline and \(m_1\) moves up the left incline.
Weight components along the slopes
Each block’s weight is resolved into a component parallel to its slope and a component perpendicular to its slope.
The parallel components are the driving terms:
Parallel components.
\[
\begin{aligned}
P_1 &= m_1g\sin\alpha \\
P_2 &= m_2g\sin\beta
\end{aligned}
\]
The right block favors motion in the positive direction, while the left block opposes that same positive direction. Therefore,
the signed driving difference is:
Driving difference.
\[
\begin{aligned}
D &= P_2-P_1 \\
&= m_2g\sin\beta - m_1g\sin\alpha
\end{aligned}
\]
If \(D>0\), the system tends to move with \(m_2\) down the right slope. If \(D<0\), it tends to move with \(m_1\) down the
left slope.
Frictionless acceleration
In the frictionless model, only the two parallel weight components drive or oppose the motion. The total moving mass is
\(m_1+m_2\), so Newton’s second law gives:
Frictionless model.
\[
\begin{aligned}
a &= \frac{m_2g\sin\beta - m_1g\sin\alpha}{m_1+m_2}
\end{aligned}
\]
Once \(a\) is known, the string tension can be computed from either block. For the left block, using the positive direction
up the left incline:
Tension from the left block.
\[
\begin{aligned}
T - m_1g\sin\alpha &= m_1a \\
T &= m_1a + m_1g\sin\alpha
\end{aligned}
\]
For the right block, using the positive direction down the right incline:
Tension from the right block.
\[
\begin{aligned}
m_2g\sin\beta - T &= m_2a \\
T &= m_2g\sin\beta - m_2a
\end{aligned}
\]
Normal forces and friction
When friction is included, the normal force on each block is needed. The normal force is the perpendicular component of
weight:
Normal forces.
\[
\begin{aligned}
N_1 &= m_1g\cos\alpha \\
N_2 &= m_2g\cos\beta
\end{aligned}
\]
Static friction can adjust up to a maximum value. The maximum static frictions are:
Maximum static friction.
\[
\begin{aligned}
f_{s1,\max} &= \mu_{s1}N_1 \\
f_{s2,\max} &= \mu_{s2}N_2
\end{aligned}
\]
A precise static-friction check can be expressed as a possible tension interval. For the left block at rest, the tension
must lie inside:
Left static interval.
\[
\begin{aligned}
T &\in [P_1-\mu_{s1}N_1,\ P_1+\mu_{s1}N_1]
\end{aligned}
\]
For the right block at rest, the tension must lie inside:
Right static interval.
\[
\begin{aligned}
T &\in [P_2-\mu_{s2}N_2,\ P_2+\mu_{s2}N_2]
\end{aligned}
\]
If these intervals overlap, static friction can hold the system at rest and \(a=0\). If they do not overlap, the blocks slide
and kinetic friction must be used.
Kinetic friction during sliding
During sliding, kinetic friction has fixed magnitudes:
Kinetic friction magnitudes.
\[
\begin{aligned}
f_{k1} &= \mu_{k1}N_1 \\
f_{k2} &= \mu_{k2}N_2
\end{aligned}
\]
Let \(s=+1\) when the motion is in the positive direction, and \(s=-1\) when the motion is reversed. Friction opposes the
motion, so the acceleration becomes:
Sliding acceleration with kinetic friction.
\[
\begin{aligned}
a &= \frac{(P_2-P_1)-s(f_{k1}+f_{k2})}{m_1+m_2}
\end{aligned}
\]
The sign of \(a\) tells the direction of the acceleration. Positive means \(m_2\) accelerates down the right incline.
Negative means \(m_1\) accelerates down the left incline.
| Case |
Main relation |
Meaning |
| Parallel component left |
\(P_1=m_1g\sin\alpha\) |
Pulls \(m_1\) down the left incline |
| Parallel component right |
\(P_2=m_2g\sin\beta\) |
Pulls \(m_2\) down the right incline |
| Frictionless acceleration |
\(a=(P_2-P_1)/(m_1+m_2)\) |
Positive means \(m_2\) moves down |
| Normal forces |
\(N_1=m_1g\cos\alpha,\quad N_2=m_2g\cos\beta\) |
Needed for friction terms |
| Static friction |
Check overlap of possible tension intervals |
If intervals overlap, \(a=0\) |
| Kinetic friction |
\(f_k=\mu_kN\) |
Used only after sliding begins |
This model assumes a light inextensible string, an ideal pulley, blocks that remain in contact with their inclines, and
friction coefficients that are constant over each surface.
