Tension in the String: It is possible to infer the tension in the string from the motion of the masses.

Tension in the String: It is possible to infer the tension in the string from the motion of the masses. Determine the expression for the tension in the string (on the left) in terms of a, m1, m2, and g. This expression may include all of these terms. Jerel 16. Again, consider the situation where m1 is given an initial downward tug. Predict how the tension in the string compares (larger, smaller, same) to the gravitational force on m1 while it is moving down? How does it compare while moving up? 17. In Trial 5, we recorded data for one run of this experiment. Use this data and the formula from part 15 to determine the tension in the string for both periods. How does the tension in the string when moving down compare to that when moving up? How do these compare to the force exerted by gravity on m1?

The Correct Answer and Explanation is:

Let’s break down the problem step by step:

Expression for Tension in the String:

Consider a two-mass system where two masses m1m_1m1​ and m2m_2m2​ are connected by a string passing over a pulley. The masses can move vertically, and we need to determine the tension TTT in the string.

• For mass m1m_1m1​: The forces acting on m1m_1m1​ are:
  • The gravitational force downward: Fg1=m1gF_{g1} = m_1 gFg1​=m1​g.
  • The tension in the string upward: TTT.
  Applying Newton’s second law to mass m1m_1m1​: F=ma⇒T−m1g=m1aF = ma \quad \Rightarrow \quad T – m_1 g = m_1 aF=ma⇒T−m1​g=m1​a Thus, the tension on mass m1m_1m1​ can be expressed as: T=m1g+m1a(Equation 1)T = m_1 g + m_1 a \quad \text{(Equation 1)}T=m1​g+m1​a(Equation 1)
• For mass m2m_2m2​: The forces acting on m2m_2m2​ are:
  • The gravitational force downward: Fg2=m2gF_{g2} = m_2 gFg2​=m2​g.
  • The tension in the string upward: TTT.
  Applying Newton’s second law to mass m2m_2m2​: F=ma⇒m2g−T=m2aF = ma \quad \Rightarrow \quad m_2 g – T = m_2 aF=ma⇒m2​g−T=m2​a Thus, the tension on mass m2m_2m2​ is: T=m2g−m2a(Equation 2)T = m_2 g – m_2 a \quad \text{(Equation 2)}T=m2​g−m2​a(Equation 2)

Inference about Tension While Moving Down:

• When m1m_1m1​ is moving downward: The downward motion implies that m1m_1m1​ is accelerating in the direction of gravity. In this case, the tension in the string will be less than the gravitational force on m1m_1m1​. This is because the gravitational force is helping the motion of m1m_1m1​, and the tension is counteracting some of that force. So, we expect: T<m1gwhen m1 is moving downward.T < m_1 g \quad \text{when \( m_1 \) is moving downward.}T<m1​gwhen m1​ is moving downward.

Inference about Tension While Moving Up:

• When m1m_1m1​ is moving upward: Now, m1m_1m1​ is moving against the direction of gravity, so the tension in the string must be greater than the gravitational force on m1m_1m1​ to overcome gravity and accelerate upwards. Thus, we expect: T>m1gwhen m1 is moving upward.T > m_1 g \quad \text{when \( m_1 \) is moving upward.}T>m1​gwhen m1​ is moving upward.

Comparison of Tension Moving Down vs. Moving Up:

Using the formulas for TTT from the two masses, we can make predictions:

• While moving downward, the tension is less than the gravitational force on m1m_1m1​, and we expect the net force on m1m_1m1​ to be smaller than the net force on m2m_2m2​. This is reflected in the equation for downward motion: T=m1g+m1aT = m_1 g + m_1 aT=m1​g+m1​a
• While moving upward, the tension will be larger than the gravitational force on m1m_1m1​, and the net force on m1m_1m1​ will be greater. This results in a higher tension in the string.

Trial 5 Data:

Once you have the data for Trial 5, you can substitute the values of m1m_1m1​, m2m_2m2​, aaa, and ggg into the equations for tension during the upward and downward motions. From the values you obtain, you can compare:

1. The tension in the string when moving down vs. up.
2. The tension in the string relative to the force exerted by gravity on m1m_1m1​.

In general, the tension will be smaller when moving down and larger when moving up compared to the gravitational force on m1m_1m1​. The magnitude of acceleration aaa will also affect how much larger or smaller the tension is.