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Chapter 14: Problem 11

The force of \(F=50 \mathrm{N}\) is applied to the cord when \(s=2 \mathrm{m} .\) If the 6 -kg collar is orginally at rest, determine its velocity at \(s=0 .\) Neglect friction.

Short Answer

Expert verified
The velocity of the 6-kg collar at \(s=0\) is approximately \(7.75 m/s\).

Step by step solution

01

Calculation of Work Done

Work done is the product of the force \(F\) and the displacement \(s\). Knowing that \(F = 50N\) and \(s = 2m\), we can calculate the work done \(W\) by multiplying these two values: \(W = F \times s = 50N \times 2m = 100J\) (Joules because work is energy and its SI unit is Joules).
02

Application of Work-Energy Theorem

According to the work-energy theorem, the work done on an object equals the change in its kinetic energy. The initial kinetic energy \(K_1\) is \(0J\) since the collar is originally at rest. The final kinetic energy \(K_2\) is the work done \(W\) since there's no friction. Therefore, \(K_2 = W = 100J\).
03

Calculation of Final Velocity

We find the final velocity \(v\) from the final kinetic energy. We know \(K_2 = (1/2) m v^2\), where \(m = 6kg\) is the mass of the collar. Solving for \(v\) from this equation, we get \(v = \sqrt{(2 \times K_2) / m} = \sqrt{(2 \times 100J) / 6kg} \approx 7.75 m/s\). This square root operation ensures that our result for velocity will be positive, which respects the rightward direction of the applied force.

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Key Concepts

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

Work Done in Physics
Work is a fundamental concept in physics and is often associated with a force causing displacement. When we talk about work done, it is in relation to how much energy is transferred when a force is applied over a distance.
In the context of our problem:
  • Force applied, \( F = 50\, \text{N} \)
  • Displacement, \( s = 2\, \text{m} \)
Work done \( W \) is given by the formula \( W = F \times s \). Here, it means we multiply the force by the distance over which it is applied, resulting in \( 100\, \text{J} \) (Joules). Joules are the units of energy, and this energy can then be used to perform various tasks, such as moving an object. In this case, the energy is used to accelerate the collar, moving it from rest to a faster state.
Kinetic Energy
Kinetic energy is the energy possessed by an object due to its motion. It is a vital part of the work-energy principle.
When the force does work on an object, it transfers energy to the object, which then moves. Kinetic energy is calculated using the formula:
  • \( K = \frac{1}{2} m v^2 \)
where:
  • \( m \) is the mass
  • \( v \) is the velocity
In our scenario, the collar initially has no kinetic energy. This is because it is at rest, meaning its velocity is zero. As work is done on the collar, its kinetic energy increases to equal the energy transferred, which is \( 100\, \text{J} \). This energy change illustrates the work-energy theorem: the work done on the object is equal to the change in its kinetic energy.
Velocity Calculation
The velocity of an object tells us how fast it is moving in a particular direction. To find the final velocity of the collar, we relate it to the kinetic energy. Given the final kinetic energy:
  • \( K_2 = 100\, \text{J} \)
We can rearrange the kinetic energy formula to solve for velocity:
  • \( v = \sqrt{\frac{2K}{m}} \)
  • For our collar, with mass \( m = 6\, \text{kg} \), this becomes \( v = \sqrt{\frac{200}{6}} \approx 7.75\, \text{m/s} \)
This calculation gives us the speed of the collar after the force has been applied. The calculation also highlights one key aspect: velocity is a direct result of how much energy is transferred to the object.
Applied Force
An applied force is any force that a person or machine exerts on an object. It can change the velocity of an object, causing it to accelerate. In mechanical systems like ours with the collar:
  • The applied force is \( 50\, \text{N} \)
  • This force acts in the direction of the movement, causing acceleration
As the collar moves, this applied force does "work," which results in a change in the kinetic energy of the collar. Without opposing forces like friction, all the energy from the applied force translates into motion, increasing the object's speed.
Mechanics Problem Solving
Solving mechanics problems often requires understanding a series of physics principles: from forces and work, to energy and motion. Here鈥檚 a simple roadmap to approach such problems:
  • Understand all given values and conditions (forces, mass, initial conditions, etc.)
  • Identify relevant physics laws (e.g., work-energy theorem)
  • Apply formulas appropriately to solve for unknowns (like velocity)
  • Check units to ensure all quantities are consistent (e.g., N, m, kg, J, m/s)
In our exercise:
  • We identified the work done using a known force and displacement
  • We applied the work-energy theorem to find the change in kinetic energy
  • We used the kinetic energy to determine the final velocity
Such structured approaches and understanding of concepts enable effective problem-solving in mechanics.

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Most popular questions from this chapter

The spring bumper is used to arrest the motion of the \(4-1 b\) block, which is sliding toward it at \(v=9 \mathrm{ft} / \mathrm{s}\). As shown, the spring is confined by the plate \(P\) and wall using cables so that its length is \(1.5 \mathrm{ft}\). If the stiffness of the spring is \(k=50\) lb/ft, determine the required unstretched length of the spring so that the plate is not displaced more than \(0.2 \mathrm{ft}\) after the block collides into it. Neglect friction, the mass of the plate and spring, and the energy loss between the plate and block during the collision.

The block has a weight of \(80 \mathrm{lb}\) and rests on the floor for which \(\mu_{k}=0.4 .\) If the motor draws in the cable at a constant rate of \(6 \mathrm{ft} / \mathrm{s}\), determine the output of the motor at the instant \(\theta=30^{\circ} .\) Neglect the mass of the cable and pulleys.

The spring has a stiffiness \(k=50 \mathrm{N} / \mathrm{m}\) and an unstretched length of \(0.3 \mathrm{m}\). If it is attached to the \(2-\mathrm{kg}\) smooth collar and the collar is released from rest at \(A\) \(\left(\theta=0^{\circ}\right),\) determine the speed of the collar when \(\theta=60^{\circ}\) The motion occurs in the horizontal plane. Neglect the size of the collar.

For protection, the barrel barrier is placed in front of the bridge pier. If the relation between the force and deflection of the barrier is \(F=\left(90\left(10^{3}\right) x^{1 / 2}\right)\) lb, where \(x\) is in ft, determine the car's maximum penetration in the barrier. The car has a weight of 4000 lb and it is traveling with a speed of \(75 \mathrm{ft} / \mathrm{s}\) just before it hits the barrier.

If the jet on the dragster supplies a constant thrust of \(T=20 \mathrm{kN}\), determine the power generated by the jet as a function of time. Neglect drag and rolling resistance, and the loss of fuel. The dragster has a mass of \(1 \mathrm{Mg}\) and starts from rest.
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