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Chapter 9: Problem 15

As a stunt motorcyclist you are trying to run through a vertical loop with radius \(3 \mathrm{~m}\). What speed do you need to have at the top of the loop in order not to fall down?

Short Answer

Expert verified
About 5.42 m/s

Step by step solution

01

Understand the forces acting at the top of the loop

At the top of the loop, the only force acting on the motorcycle is gravity (weight), which must provide the necessary centripetal force to keep the motorcycle moving in a circular path.
02

Apply the condition for minimum speed

The condition for the minimum speed at the top of the loop is that the gravitational force (mg) must be equal to the centripetal force needed to keep the motorcycle in circular motion, which can be expressed as: \[ mg = \frac{mv^2}{r} \]
03

Solve for speed

Given that the radii of the loop (r) is 3 meters and the gravitational acceleration (g) is approximately 9.8 m/s², substitute these values into the equation: \[ mg = \frac{mv^2}{3} \]By simplifying, we get: \[ g = \frac{v^2}{3} \]Solving for v gives: \[ v^2 = 3g \]Finally, take the square root to find the speed: \[ v = \frac{\text{0} + \text{SqaureRoot}} \]\[ v = \sqrt{3 \times 9.8} \] \[ v \approx 5.42 \text{ m/s} \]

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Key Concepts

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

Gravitational Force
Gravitational force is one of the fundamental forces of nature. It is the force that pulls objects toward the center of the Earth and keeps us grounded. This force is described by the equation \( F = mg \), where \( F \) is the force, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity, approximately \( 9.8 \text{ m/s}^2 \).
Minimum Speed
When discussing the minimum speed required to stay in a curved path, we refer to the point where gravitational force alone provides the required centripetal force.
Circular Motion
Circular motion refers to an object moving along the boundary of a circle. This type of motion is influenced by a centripetal force, which is always directed towards the center of the circle.

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Most popular questions from this chapter

Walking on ice is usually more tiring than walking on a dry road. Why?

In this project we will study a phenomenon called stickslip friction. If you pull a block along a flat table with a soft spring, you will find that the block does not move continuously with a constant velocity, instead it moves in small jumps. This intermittent motion is called stick-slip friction, and it is the origin of the high-frequency vibrating tone you often hear from wheels that are not well lubricated. It is also one of the basic mechanisms leading to the wide distribution of earthquake sizes. Here, we will introduce and study a model for stick-slip friction for a block pulled by a spring sliding over a flat, horizontal surface, as illustrated in Fig. \(9.18\). The block has mass \(m\). A massless spring (with spring constant \(k\) and equilibrium length \(b\) ) is attached to the block at the point \(x\). The free end (the right-hand end in Fig. 9.18) of the spring is at the point \(x_{b}\). We move \(x_{b}\), the free end of the spring, with a constant velocity \(u\). The static and dynamic coefficients of friction for the contact between the block and the bottom surface are \(\mu_{s}\) and \(\mu_{d}\) respectively. The acceleration of gravity is \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\). The block starts at the position \(x\left(t_{0}\right)=0\) at the time \(t_{0}=0\). The position \(x_{b}\) of the free end of the spring is \(x_{b}\left(t_{0}\right)=x\left(t_{0}\right)+b\) at \(t_{0}\). (a) Draw a free-body diagram for the block. (b) Find the position of the spring attachment point \(x_{b}(t)\) as a function of time. (c) Show that the force, \(\mathbf{F}\) on the block from the spring is \(\mathbf{F}=k\left(x_{b}-x-b\right) \mathbf{i}\). Stationary state: First, let us characterize the stationary state, where the block is moving at a constant velocity. (d) Identify the forces acting on the block and draw a free-body diagram for the block in the stationary state. (e) Introduce force models for all the forces acting on the block. Find the normal force, \(N\), on the block. (f) Find the acceleration of the block in the stationary state. (g) Find the elongation \(\Delta L\) of the spring in the stationary state. (h) Find the position \(x(t)\) of the block as a function of time in the stationary state. Starting from rest: Let us now address the situation where the block starts from rest. That is, we assume that the block starts at \(x\left(t_{0}\right)=0 \mathrm{~m}\) with \(v\left(t_{0}\right)=0 \mathrm{~m} / \mathrm{s}\) at the time \(t_{0}=0 \mathrm{~s}\). (i) Identify the force acting on the block and draw a free-body diagram of the block before the block starts moving. Introduce force models for all the forces. (j) Find the elongation \(\Delta L\) of the spring at the instant the block starts moving. (k) Assume that the block starts at rest. Find the friction force on the block as a function of time in the period before the block starts moving. Sketch the friction force as a function of time until some time after the block has started moving. (1) Show that the acceleration of the block immediately after it starts moving is \(a=(k / m)\left(x_{b}-x-b\right)-\mu_{d} g\). Explain why you cannot use this relation for the acceleration to determine the subsequent motion of the block. General motion: Now, we will develop a general method to find the motion, \(x(t)\), of the block. First, we study the case when \(u=0 \mathrm{~m} / \mathrm{s}\) and the coefficients of friction are zero, \(\mu_{s}=\mu_{d}=0\). (m) Find an expression for the horizontal acceleration of the block. Show that \(x(t)=\) \(\left(v_{0} / \omega\right) \sin \omega t\), where \(\omega=(k / m)^{1 / 2}\), when \(v(0)=v_{0} .\) (n) Write a numerical algorithm to find the position and velocity of the block at a time \(t_{i}+\Delta t, x\left(t_{i}+\Delta t\right)\) and \(v\left(t_{i}+\Delta t\right)\), given the position and velocity of the block at a time \(t_{i}, x\left(t_{i}\right)\) and \(v\left(t_{i}\right)\). (o) Implement the numerical algorithm in a program to find the position of the block as a function of time for \(m=0.1 \mathrm{~kg}, k=100 \mathrm{~N} / \mathrm{m}, b=0.1 \mathrm{~m}\) and \(v_{0}=0.1 \mathrm{~m} / \mathrm{s}\). Plot the behavior for a simulation of \(2 \mathrm{~s}\), and compare the result of your program with exact solution. Ensure that you choose a time-step \(\Delta t\) the reproduces the exact solution with sufficient accuracy. What happens if you choose a too large time-step \(\Delta t ?\) Let us now address the situation when the block is pulled at a finite velocity, \(u\). (p) Modify your program to find the position of the block when \(u=0.1 \mathrm{~m} / \mathrm{s}\) and the block starts at rest. In this case, the exact solution is: $$ x(t)=u t-\frac{u}{\omega} \sin \omega t . $$ Compare your result with the exact solution by plotting both the simulated \(x\) and the exact \(x\) in the same plot. General motion with friction: Finally, we address the full complexity of the situation, and introduce non-zero friction forces. (q) Modify your program to include friction using \(\mu_{s}=0.6, \mu_{d}=0.3\). Show a plot of \(x(t)\) for \(m=0.1 \mathrm{~kg}\) and for \(m=1.0 \mathrm{~kg}\). (r) Plot the spring force \(F\) on the block as a function of time for both values of \(m\) and explain the differences. (s) What happens if you instead decrease \(k\) to \(k=10 \mathrm{~N}\) for \(m=0.1 \mathrm{~kg}\). Can you explain the behavior?

A homogeneous rope with mass \(m\) hangs between two equally high poles. The angle between the rope and the horizontal at each of the attachment points is \(\alpha\). (a) Find the tension at each end of the rope. (b) Find the tension at the lowest point of the rope. (c) Is it possible to tighten the rope so much that \(\alpha=0\) ?

Two books are lying on top of each other on a table. The upper book has a mass \(m_{1}\), and the lower book has a mass \(m_{2}\). The coefficient of static friction between the books is \(\mu_{1}\). The coefficient of static friction between the book and the table is \(\mu_{2}\) and the coefficient of dynamic friction between the book and the table is \(\mu_{d}\). You pull on the lower book with a horizontal force \(F\). (a) How large must \(F\) be for you to start pulling both books along the table. (b) How large must \(F\) be for you to pull out only the lower book?

In this project you will learn to use Newton's laws and the force model for air resistance in a wind field to address the motion of a light object in strong winds. We start from a simple model without wind and gradually add complexity to the model, until we finally address the motion in a tornado. Motion without wind: First, we address the motion of the feather without wind. (a) Identify the forces acting on a feather while it is falling and draw a free-body diagram for the feather. (b) Introduce quantitative force models for the forces, and find an expression for the acceleration of the feather. You may assume a quadratic law for air resistance. (c) If you release the feather from rest, its velocity will tend asymptotically toward the terminal velocity, \(v_{T}\). Show that the terminal velocity is \(v_{T}=-(m g / D)^{1 / 2}\), where \(D\) is the constant in the air resistance model. (d) We release the feather from a distance \(h\) above the floor and measure the time \(t\) until the feather hits the floor. You may assume that the feather falls with a constant velocity equal to the terminal velocity. Show how you can determine \(D / m g\) by measuring the time \(t\). Estimate \(D / m g\) when you release the feather from a height of \(2.4 \mathrm{~m}\) above the floor and it takes \(4.8 \mathrm{~s}\) until it hits the floor. (e) We will now develop a more precise model where we do not assume that the velocity is constant. You release the feather from the height \(h\) at the time \(t=0 \mathrm{~s}\). Find the equation you have to solve to find the position of the feather as a function of time. What are the initial conditions? (f) Write a program that solves this equation to find the velocity and position as a function of time \(t\). Use the parameters you determined above, and test the program by ensuring that it produces the correct terminal velocity. (g) Fig. \(9.19\) shows the position and velocity calculated with the program using the parameters found above. Was the approximation in part (d) reasonable? Explain your answer. Model with wind: We have now found a model that can be used to find the motion of the feather. We will now find the motion of the feather in three dimensions while it is blowing. The velocity of the wind varies in space, so that the wind velocity \(\mathbf{w}\) is a function of the position \(\mathbf{r}\). We write this as \(\mathbf{w}=\mathbf{w}(\mathbf{r})\). (h) Find an expression for the acceleration of the feather. The expression may include the wind velocity \(\mathbf{w}(\mathbf{r})\). Let the z-axis correspond to the vertical direction. (i) Assume that the feather is moving in an approximately horizontal plane- that is you may assume that the vertical acceleration is negligible. How does the wind have
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