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Chapter 13: Problem 50

Block \(A\) has a mass \(m_{A}\) and is attached to a spring having a stiffness \(k\) and unstretched length \(l_{0}\). If another block \(B\), having a mass \(m_{B}\), is pressed against \(A\) so that the spring deforms a distance \(d\), show that for separation to occur it is necessary that \(d>2 \mu_{k} g\left(m_{A}+m_{B}\right) / k,\) where \(\mu_{k}\) is the coefficient of kinetic friction between the blocks and the ground. Also, what is the distance the blocks slide on the surface before they separate?

Short Answer

Expert verified
For the blocks to separate, the spring deformation \(d\), must be greater than \(2 \mu_{k} g\left(m_{A}+m_{B}\right) / k\). The blocks will slide a distance equal to this \(d\) before they separate.

Step by step solution

01

Identify the forces acting on the blocks

Here, two main forces act on the blocks. They are the force due to the spring compression pushing the blocks apart, \(F_{s}\) and the kinetic frictional force, \(F_{k}\), attempting to hold them together. The force due to the spring can be calculated using Hooke's law: \(F_{s} = kd\). The kinetic frictional force is given by: \(F_{k} = \mu_{k} (m_A + m_B) g\).
02

Determine the condition for separation

When there is separation, the force exerted by the spring overcomes the kinetic friction i.e., \(F_{s} > F_{k}\). This can be expressed as: \(kd > \mu_{k} (m_A + m_B) g\). From this equation, we can isolate \(d\) to get: \(d > \mu_{k} (m_A + m_B) g / k\).
03

Calculate distance before separation

The distance the blocks slide on the surface before they separate is equal to \(d\), the deformation of the spring. Therefore, by determining \(d\) from the equation in step 2, the distance the blocks slide before they separate can be obtained.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Friction
In the study of spring dynamics, addressing friction forces is crucial for understanding block movements on a surface. Kinetic friction opposes the direction of motion and is critical when two surfaces slide against each other. When a block moves over another, kinetic friction acts to resist this motion.
The magnitude of kinetic friction ($$F_k$$) can be determined by the formula:
  • \(F_k = \mu_k (m_A + m_B) g\)
Here,
  • \(\mu_k\) represents the coefficient of kinetic friction.
  • \(m_A + m_B\) is the total mass of both blocks.
  • \(g\) is the acceleration due to gravity.

Understanding this concept helps us evaluate when the spring force will overcome kinetic friction, causing separation of the blocks.
Hooke's Law
An essential concept in spring dynamics is Hooke's Law, which defines the relationship between the force exerted by a spring and its deformation. Hooke's Law states that the force exerted by a spring is directly proportional to its extension or compression.
This can be expressed mathematically as:
  • $F_s = kd$
Where:
  • $F_s$ is the force exerted by the spring.
  • $k$ is the spring constant or stiffness.
  • $d$ is the amount of deformation.

In our problem, this force works to push the blocks apart, opposing the kinetic friction. For the blocks to separate, the spring force must exceed the frictional force opposing it.
Force Analysis
Force analysis helps in understanding how various forces interact and influence motion. In the exercise, analyzing forces allows us to determine when separation occurs between the blocks.
Initially:
  • Two forces are involved: the spring force \(F_s\) and kinetic friction \(F_k\).
During separation:
  • The condition \(F_s > F_k\) must be satisfied.
This leads us to derive the equation:$$kd > \mu_k (m_A + m_B) g$$
Solving for \(d\), we find the critical deformation needed for separation:$$d > \frac{2\mu_k g (m_A + m_B)}{k}$$
Have these forces clearly defined and analyzed allows us to predict the motion of the blocks and ensure understanding of the requirements for their separation.

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Most popular questions from this chapter

A \(0.2-\mathrm{kg}\) spool slides down along a smooth rod. If the rod has a constant angular rate of rotation \(\dot{\theta}=2 \mathrm{rad} / \mathrm{s}\) in the vertical plane, show that the equations of motion for the spool are \(\ddot{r}-4 r-9.81 \sin \theta=0\) and \(0.8 \dot{r}+N_{s}-1.962 \cos \theta=0,\) where \(N_{s}\) is the magnitude of the normal force of the rod on the spool. Using the methods of differential equations, it can be shown that the solution of the first of these equations is \(r=C_{1} e^{-2 t}+C_{2} e^{2 t}-(9.81 / 8) \sin 2 t .\) If \(r, \dot{r},\) and \(\theta\) are zero when \(t=0,\) evaluate the constants \(C_{1}\) and \(C_{2}\) to determine \(r\) at the instant \(\theta=\pi / 4 \mathrm{rad}\).

Using a forked rod, a \(0.5-\mathrm{kg}\) smooth peg \(P\) is forced to move along the vertical slotted path \(r=(0.5 \theta) \mathrm{m}\), where \(\theta\) is in radians. If the angular position of the arm is \(\theta=\left(\frac{\pi}{8} t^{2}\right) \mathrm{rad},\) where \(t\) is in seconds, determine the force of the rod on the peg and the normal force of the slot on the peg at the instant \(t=2 \mathrm{~s}\). The peg is in contact with only one edge of the rod and slot at any instant.

Rod \(O A\) rotates counterclockwise at a constant angular rate \(\dot{\theta}=4 \mathrm{rad} / \mathrm{s}\). The double collar \(B\) is pin-connected together such that one collar slides over the rotating rod and the other collar slides over the circular rod described by the equation \(r=(1.6 \cos \theta) \mathrm{m}\). If both collars have a mass of \(0.5 \mathrm{~kg}\), determine the force which the circular rod exerts on one of the collars and the force that \(O A\) exerts on the other collar at the instant \(\theta=45^{\circ}\). Motion is in the horizontal plane.

A car of mass \(m\) is traveling at a slow velocity \(v_{0}\) If it is subjected to the drag resistance of the wind, which is proportional to its velocity, i.e., \(F_{D}=k v\), determine the distance and the time the car will travel before its velocity becomes \(0.5 v_{0}\). Assume no other frictional forces act on the car.

Block \(A\) has a mass \(m_{A}\) and is attached to a spring having a stiffness \(k\) and unstretched length \(l_{0}\). If another block \(B\), having a mass \(m_{B}\), is pressed against \(A\) so that the spring deforms a distance \(d,\) determine the distance both blocks slide on the smooth surface before they begin to separate. What is their velocity at this instant?
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