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Chapter 4: Problem 144

A toy cart of mass \(m_{1}\) moves on frictionless wheels as it is pulled by a string under tension \(T .\) A block of mass \(m_{2}\) rests on top of the cart. The coefficient of static friction between the cart and the block is \(\mu .\) Find the maximum tension \(T\) that will not cause the block to slide on the cart if the cart rolls on (a) a horizontal surface; (b) up a ramp of angle \(\theta\) above the horizontal. In both cases, the string is parallel to the surface on which the cart rolls.

Short Answer

Expert verified
Answer: The maximum tensions in the given scenarios are (a) on a horizontal surface - \(T_{max} = \mu m_{2}g\) and (b) on an inclined ramp at an angle θ - \(T_{max} = m_{2}g\sin(\theta) + \mu m_{2}g\cos(\theta)\).

Step by step solution

01

Analyze the forces on the block

Since the block is at rest on the cart, the force of gravity (\(m_{2}g\)) acts downward, and the static friction force between the block and the cart (\(f_{s}\)) acts in the opposite direction to the force exerted by the tension \(T\). The normal force also acts on the block, but as there is no vertical motion, it is equal to the gravitational force acting on the block.
02

Set up the equations for the forces

As the block is in equilibrium, the net force acting on it is zero. This can be written as: \(f_{s} = m_{2}a_{2}\), where \(a_{2}\) is the acceleration of the block.
03

Calculate the maximum static friction force

The maximum static friction force that can act on the block before it starts sliding is given by \(f_{s\_max} = \mu N\), where \(\mu\) is the coefficient of static friction and \(N\) is the normal force. Since \(N = m_{2}g\), we have \(f_{s\_max} = \mu m_{2}g\).
04

Find the maximum tension

For the block to not slide on the cart, the tension force must not exceed the maximum static friction force; therefore, we have \(T \leq f_{s\_max}\). This means that the maximum tension in this scenario is \(T_{max} = \mu m_{2}g\). ##Inclined Ramp##
05

Analyze the forces on the block for an inclined ramp

Now, the cart is on an inclined ramp at an angle \(\theta\). The gravitational force \(m_{2}g\) has two components: \(m_{2}g\cos(\theta)\), perpendicular to the ramp, and \(m_{2}g\sin(\theta)\), parallel to the ramp, down the incline. We also have the normal force \(N\) and the static friction force \(f_s\).
06

Set up the equations for the forces on the inclined ramp

For the block to be in equilibrium and not slide on the cart, \(T = m_{2}g\sin(\theta) + f_{s}\).
07

Calculate the maximum static friction force on the inclined ramp

The maximum static friction force can be found as \(f_{s\_max} = \mu N\). In this case, since \(N = m_{2}g\cos(\theta)\), we have \(f_{s\_max} = \mu m_{2}g\cos(\theta)\).
08

Find the maximum tension for the inclined ramp

For the block to not slide on the cart, the tension force must not exceed the maximum static friction force, just like in the case of a horizontal surface. Therefore, we must have \(T \leq m_{2}g\sin(\theta) + f_{s\_max}\). Thus, the maximum tension in this scenario is \(T_{max} = m_{2}g\sin(\theta) + \mu m_{2}g\cos(\theta)\).

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Most popular questions from this chapter

A crate of potatoes of mass \(18.0 \mathrm{kg}\) is on a ramp with angle of incline \(30^{\circ}\) to the horizontal. The coefficients of friction are \(\mu_{\mathrm{s}}=0.75\) and \(\mu_{\mathrm{k}}=0.40 .\) Find the frictional force (magnitude and direction) on the crate if the crate is at rest.
A 2.0 -kg toy locomotive is pulling a 1.0 -kg caboose. The frictional force of the track on the caboose is \(0.50 \mathrm{N}\) backward along the track. If the train's acceleration forward is \(3.0 \mathrm{m} / \mathrm{s}^{2},\) what is the magnitude of the force exerted by the locomotive on the caboose?
A refrigerator magnet weighing \(0.14 \mathrm{N}\) is used to hold up a photograph weighing \(0.030 \mathrm{N}\). The magnet attracts the refrigerator door with a magnetic force of 2.10 N. (a) Identify the interactions between the magnet and other objects. (b) Draw an FBD for the magnet, showing all the forces that act on it. (c) Which of these forces are long-range and which are contact forces? (d) Find the magnitudes of all the forces acting on the magnet.
A woman who weighs \(600 \mathrm{N}\) sits on a chair with her feet on the floor and her arms resting on the chair's armrests. The chair weighs $100 \mathrm{N}\(. Each armrest exerts an upward force of \)25 \mathrm{N}$ on her arms. The seat of the chair exerts an upward force of \(500 \mathrm{N}\). (a) What force does the floor exert on her feet? (b) What force does the floor exert on the chair? (c) Consider the woman and the chair to be a single system. Draw an FBD for this system that includes all of the external forces acting on it.
What is the acceleration of an automobile of mass $1.40 \times 10^{3} \mathrm{kg}\( when it is subjected to a forward force of \)3.36 \times 10^{3} \mathrm{N} ?$
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