/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 48 The smooth block \(B\) of neglig... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 48

The smooth block \(B\) of negligible size has a mass \(m\) and rests on the horizontal plane. If the board \(A C\) pushes on the block at an angle \(\theta\) with a constant acceleration \(\mathbf{a}_{0}\), determine the velocity of the block along the board and the distance \(s\) the block moves along the board as a function of time \(t\). The block starts from rest when \(s=0, t=0\). Prob.

Short Answer

Expert verified
The velocity \( v \) of the block along the board as a function of time \( t \) is given by \( v= a_0 \sin(\theta) t \). The distance \( s \) the block moves along the board as a function of time \( t \) is given by \( s= 0.5 a_0 \sin(\theta) t^2 \).

Step by step solution

01

Analyze Forces on the Block

Identify the forces acting on the block. There's the weight of the block acting downwards, which is \(mg\), and the normal force acting perpendicular to (and upwards along) the board, \(N\). The force due to acceleration is acting along the board and that is \(ma_0\). Decompose these forces into components along the board and perpendicular to the board.
02

Apply Newton's Second Law

In the direction perpendicular to the motion of the block (along line AC), the net force is zero. This gives us the equation: \( N = mg\cos(\theta) \). In the direction of the motion of the block (along line AC), the net force is not zero (equals to mass times net acceleration in that direction). This gives us the equation: \( m a = ma_0\sin(\theta) \). Solve for the acceleration \(a\) to get \( a=a_0\sin(\theta)\).
03

Apply Kinematic Equations

We know the initial velocity \( v_0 = 0 \) (from the prompt, as the block starts from rest), and acceleration \( a = a_0 \sin(\theta)\). Using the kinematic equations \(v = v_0 + a t\) and \(s = v_0 t + 0.5 a t^2 \), we can solve for \(v\) and \(s\) as functions of \(t\). This gives \( v = a_0 \sin(\theta) t\) and \( s = 0.5 a_0\sin(\theta) t^2\).

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Key Concepts

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

Newton's Second Law
Newton's Second Law is about understanding how forces cause objects to move. It is fundamental to dynamics and is expressed with the formula \( F = ma \), where \( F \) represents the net force applied to an object, \( m \) is its mass, and \( a \) is the acceleration produced.
In our exercise, this law helps to determine how forces along the board affect the block. The block experiences an acceleration \( a_0 \), applied at an angle \( \theta \), different from its weight. Here we break forces into components to apply Newton's Second Law effectively.
  • The force along the board causes the block to accelerate, calculated as \( ma_0\sin(\theta) \).
  • Perpendicularly, forces balance each other with \( N = mg\cos(\theta) \), meaning the block doesn't move vertically.
This decomposition of forces ensures that the net force is correctly applied in the moving direction, allowing us to apply the kinematic equations with the correct acceleration.
Kinematic Equations
Kinematic equations help describe motion, providing relationships between velocity, acceleration, time, and displacement without considering forces directly.
The block starts with an initial velocity \( v_0 = 0 \) as it begins from rest. With the acceleration \( a = a_0 \sin(\theta) \) found from Newton's Second Law, we use these equations:
  • The velocity as a function of time is \( v = v_0 + at \), simplifying to \( v = a_0 \sin(\theta) t \), since \( v_0 = 0 \).
  • For displacement, the equation \( s = v_0 t + 0.5 a t^2 \) applies, leading to \( s = 0.5 a_0\sin(\theta) t^2 \).
With these equations, we can predict how fast the block will move and how far it will travel over time, using the given initial conditions and forces.
Force Analysis
Force analysis involves breaking down the forces on an object to understand its motion under those forces.
For the block on the board, the following forces were considered:
  • Gravitational Force \( mg \): Acts downward, pulling the block towards the Earth.
  • Normal Force \( N \): Acts perpendicular to the board's surface, countering the gravitational pull without resulting in vertical movement.
  • Force due to board's acceleration \( ma_0 \): Causes the block to slide along the board.
We analyze these forces by decomposing them according to their directions:
  • Along the board, \( ma_0 \sin(\theta) \) drives the motion.
  • Perpendicular to the board, \( mg \cos(\theta) \) and \( N \) balance each other, preventing vertical motion.
Effective force analysis simplifies problems allowing kinetic and dynamic equations to solve for velocities and positions.

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Most popular questions from this chapter

Determine the maximum constant speed at which the pilot can travel around the vertical curve having a radius of curvature \(\rho=800 \mathrm{~m}\), so that he experiences a maximum acceleration \(a_{n}=8 g=78.5 \mathrm{~m} / \mathrm{s}^{2} .\) If he has a mass of \(70 \mathrm{~kg}\) determine the normal force he exerts on the seat of the airplane when the plane is traveling at this speed and is at its lowest point.

The vehicle is designed to combine the feel of a motorcycle with the comfort and safety of an automobile. If the vehicle is traveling at a constant speed of \(80 \mathrm{~km} / \mathrm{h}\) along a circular curved road of radius \(100 \mathrm{~m}\), determine the tilt angle \(\theta\) of the vehicle so that only a normal force from the seat acts on the driver. Neglect the size of the driver. Prob.

The airplane, traveling at a constant speed of \(50 \mathrm{~m} / \mathrm{s}\), is executing a horizontal turn. If the plane is banked at \(\theta=15^{\circ}\), when the pilot experiences only a normal force on the seat of the plane, determine the radius of curvature \(\rho\) of the turn. Also, what is the normal force of the seat on the pilot if he has a mass of \(70 \mathrm{~kg}\). rob. \(13-79\)

The collar \(A\), having a mass of \(0.75 \mathrm{~kg}\), is attached to a spring having a stiffness of \(k=200 \mathrm{~N} / \mathrm{m}\). When \(\operatorname{rod} B C\) rotates about the vertical axis, the collar slides outward along the smooth \(\operatorname{rod} D E\). If the spring is unstretched when \(s=0\), determine the constant speed of the collar in order that \(s=100 \mathrm{~mm}\). Also, what is the normal force of the rod on the collar? Neglect the size of the collar. Prob. 13-57

A girl, having a mass of \(15 \mathrm{~kg}\), sits motionless relative to the surface of a horizontal platform at a distance of \(r=5 \mathrm{~m}\) from the platform's center. If the angular motion of the platform is slowly increased so that the girl's tangential component of acceleration can be neglected, determine the maximum speed which the girl will have before she begins to slip off the platform. The coefficient of static friction between the girl and the platform is \(\mu=0.2\). Prob. \(13-52\)
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