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Chapter 17: Problem 72

A 9-in.-radius cylinder of weight 18 lb rests on a 6-lb carriage. The system is at rest when a force P of magnitude 2.5 lb is applied as shown for 1.2 s. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine the resulting velocity of (a) the carriage, (b) the center of the cylinder.

Short Answer

Expert verified
Both the carriage and the cylinder's center reach a velocity of 4.026 ft/s.

Step by step solution

01

Identify Given Data and Concepts

The problem involves a cylinder and a carriage. The cylinder has a radius of 9 inches and weighs 18 lb, while the carriage itself weighs 6 lb. A horizontal force \( P \) of 2.5 lb is applied for 1.2 seconds. We need to calculate the velocity of the carriage and the cylinder's center after this force is applied, knowing that the cylinder rolls without slipping.
02

Calculate the Total Mass

The total mass of the system is the combined mass of the cylinder and the carriage. Convert the weight of each component to mass using the equation \( m = \frac{W}{g} \), where \( g \) is typically 32.2 ft/s虏 (gravity acceleration in imperial units). For the cylinder: \( m_c = \frac{18}{32.2} \approx 0.559 \) slug. For the carriage: \( m_{ca} = \frac{6}{32.2} \approx 0.186 \) slug. The total mass \( m_t = m_c + m_{ca} = 0.559 + 0.186 = 0.745 \) slug.
03

Apply Newton's Second Law to the System

Since the cylinder rolls without slipping, both the cylinder and the carriage move together under the force. Use the equation \( F = ma \) to calculate the acceleration \( a \) generated by the force applied to the system.\[ a = \frac{P}{m_t} = \frac{2.5}{0.745} \approx 3.355 \text{ ft/s}^2 \]
04

Calculate the Velocity of the Carriage

The velocity \( v \) can be found using the equation of motion \( v = u + at \), where initial velocity \( u = 0 \) as the system is initially at rest.\[ v = 0 + (3.355 \times 1.2) \approx 4.026 \text{ ft/s} \]Thus, the velocity of the carriage after the force is applied is approximately 4.026 ft/s.
05

Determine the Velocity of the Cylinder's Center

For rolling motion without sliding, the linear velocity of the cylinder's center \( v_c \) is equal to the velocity of the carriage, since they move together. Therefore, the velocity of the cylinder's center after the force is applied is also 4.026 ft/s.

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Key Concepts

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

Newton's Second Law
To understand the motion of objects, Newton's Second Law comes in handy. This law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, it is expressed as \( F = ma \). In our exercise, the force \( P \) applied to the system of the cylinder and carriage causes both to accelerate together.

Here's how Newton's Second Law relates to the problem at hand:
  • We first calculate the total mass of the system by converting individual weights to masses.
  • With the force known, we can find the acceleration using \( a = \frac{P}{m_t} \).
  • This tells us how fast the velocity of the system changes over time.
Ultimately, this allows us to determine how quickly the entire setup starts moving once the force is applied.
Conservation of Momentum
Conservation of momentum is a core principle in dynamics, stating that the total momentum of a closed system remains constant if no external forces are acting on it. In the problem you worked on, things are a bit unique because the cylinder is rolling without slipping on the carriage, which means they move as a single system.

Even though an external force is applied here, conservation principles guide our understanding:
  • The mass times velocity (momentum) for any system is crucial in analyzing dynamic problems.
  • Once we calculate the final velocity, it represents how momentum was conserved between the two components - the carriage and the cylinder.
  • So, the system's momentum before the force was applied was zero, and it matches the momentum calculated using the velocity from step 4.
In short, while external forces can change an object's motion, the idea of momentum conservation helps track those changes.
Rolling Motion
Rolling motion is a type of motion experienced by an object when it rolls over a surface. It's important to note the distinction between rolling without slipping and sliding. For rolling without slipping to occur, the speed at the point of contact with the surface must instantaneously be zero, ensuring frictional force prevents slipping.

In the exercise, the cylinder rolls without slipping on top of the carriage. This means:
  • The velocity of the carriage at any moment equals the linear velocity of the cylinder's center.
  • This relationship simplifies the calculation of the cylinder鈥檚 motion to just finding the carriage's velocity.
  • In mechanical terms, rolling without slipping involves dealing with both rotational motion and translational motion.
Understanding these aspects of rolling helps describe how different forces and motions are interconnected within dynamic systems.

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Most popular questions from this chapter

Sphere \(A\) of mass \(m_{A}=2 \mathrm{kg}\) and radius \(r=40 \mathrm{mm}\) rolls without slipping with a velocity \(\bar{v}_{1}=2 \mathrm{m} / \mathrm{s}\) on a horizontal surface when it hits squarely a uniform slender bar \(B\) of mass \(m_{B}=0.5 \mathrm{kg}\) and length \(L=100 \mathrm{mm}\) that is standing on end and is at rest. Denoting by \(\mu_{k}\) the coefficient of kinetic friction between the sphere and the horizontal surface, neglecting friction between the sphere and the bar, and knowing the coefficient of restitution between A and B is 0.1, determine the angular velocities of the sphere and the bar immediately after the impact.

The uniform \(4-\mathrm{kg}\) cylinder \(A\) with a radius of \(r=150 \mathrm{mm}\) has an angular velocity of \(\omega_{0}=50 \mathrm{rad} / \mathrm{s}\) when it is brought into contact with an identical cylinder \(B\) that is at rest. The coefficient of kinetic friction at the contact point \(D\) is \(\mu_{k}\). After a period of slipping, the cylinders attain constant angular velocities of equal magnitude and opposite direction at the same time. Knowing that cylinder \(A\) executes three revolutions before it attains a constant angular velocity and cylinder \(B\) executes one revolution before it attains a constant angular velocity, determine (a) the final angular velocity of each cylinder, \((b)\) the coefficient of kinetic friction \(\mu_{k} .\)

A small grinding wheel is attached to the shaft of an electric motor that has a rated speed of 3600 rpm. When the power is turned off, the unit coasts to rest in 70 s. The grinding wheel and rotor have a combined weight of 6 lb and a combined radius of gyration of 2 in. Determine the average magnitude of the couple due to kinetic friction in the bearings of the motor.

The drive belt on a vintage sander transmits \(1 / 2\) hp to a pulley that has a diameter of \(d=4\) in. Knowing that the pulley rotates at 1450 rpm, determine the tension difference \(T_{1}-T_{2}\) between the tight and slack sides of the belt.

The circular platform A is fitted with a rim of 200-mm inner radius and can rotate freely about the vertical shaft. It is known that the platform-rim unit has a mass of 5 kg and a radius of gyration of 175 mm with respect to the shaft. At a time when the platform is rotating with an angular velocity of 50 rpm, a 3-kg disk B of radius 80 mm is placed on the platform with no velocity. Knowing that disk B then slides until it comes to rest relative to the platform against the rim, determine the final angular velocity of the platform.
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