i asked this question earlier but didnt really get how to find the tension equation.
Two blocks are connected by a string as shown. The inclination of the ramp is θ = 36° while the masses of the blocks are m1 = 3.7 kg and m2 = 16.2 kg. Friction is negligible.
(search question above for diagram)
Write an equation for the tension in the string in terms of m2, the acceleration due to gravity g, and the acceleration of the two blocks a.
What is the tension in the rope in newtons?
To find the tension equation, we need to analyze the forces acting on the two blocks connected by the string. Assuming m1 is on the inclined ramp and m2 is hanging vertically, the forces involved are gravity, normal force, and tension.
For m1 on the inclined ramp:
- The force of gravity, mg, can be resolved into two components: the component parallel to the ramp, mg sin(θ), and the component perpendicular to the ramp, mg cos(θ).
- The normal force, N, is the force exerted by the inclined plane on m1 perpendicular to the ramp surface.
- The tension, T, is the force transmitted by the string connecting m1 and m2 and acts parallel to the ramp.
For m2 hanging vertically:
- The force of gravity, m2 * g, acts directly downward.
- The tension, T, is transmitted by the string and acts upward.
To start, let's consider m1 on the inclined ramp.
1. Apply Newton's second law, F = ma, to the forces acting parallel to the ramp:
T - mg sin(θ) = m1 * a
Next, let's consider m2 hanging vertically.
2. Apply Newton's second law, F = ma, to the forces acting on m2:
T - m2 * g = m2 * a
Now, we have two equations (1 and 2) relating the tension, T, to the masses, accelerations, and gravitational acceleration.
To eliminate the acceleration, a, we can solve for it in terms of m1, m2, and g using equation 1:
a = (T - mg sin(θ)) / m1
Substituting this expression for a into equation 2, we can eliminate the acceleration:
T - m2 * g = m2 * ((T - mg sin(θ)) / m1)
Next, we rearrange the equation to isolate T:
T - (m2 * T / m1) + m2 * g = m2 * g sin(θ)
T - (m2 * T / m1) = m2 * g (sin(θ) - 1)
T * (1 - (m2 / m1)) = m2 * g (sin(θ) - 1)
T = (m2 * g * (sin(θ) - 1)) / (1 - (m2 / m1))
Now that we have the tension equation, we can substitute the given values:
T = (16.2 kg * 9.8 m/s² * (sin(36°) - 1)) / (1 - (16.2 kg / 3.7 kg))
Evaluating this equation will give us the tension in the rope.