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Chapter 8: Problem 61

A solid sphere of mass 0.600 kg rolls without slipping along a horizontal surface with a transnational speed of \(5.00 \mathrm{m} / \mathrm{s} .\) It comes to an incline that makes an angle of \(30^{\circ}\) with the horizontal surface. Ignoring energy losses due to friction, to what vertical height above the horizontal surface does the sphere rise on the incline?

Short Answer

Expert verified
Answer: To calculate the vertical height to which the sphere rises along the incline, follow the steps: 1. Calculate the initial total kinetic energy of the sphere. 2. Calculate the gravitational potential energy at the highest point on the incline. 3. Use the conservation of mechanical energy principle to equate the initial total kinetic energy to the gravitational potential energy at the highest point. 4. Solve for the vertical height (h) using the given values for mass, speed, and acceleration due to gravity. 5. Calculate the distance along the inclined plane using the calculated value of h and the given angle of inclination.

Step by step solution

01

Calculate the initial kinetic energy of the sphere

To do this, we need to find the total kinetic energy of the sphere, which consists of translational kinetic energy and rotational kinetic energy. For a sphere, the moment of inertia \(I = \frac{2}{5}mr^2\). The total kinetic energy, KE, is the sum of translational and rotational kinetic energy and can be found using the formula: KE_total = \(\frac{1}{2} mv^2 + \frac{1}{2} Iω^2\) Since the sphere is rolling without slipping, we can use the relation \(v = rω\). Using the given data, \(m = 0.600 \mathrm{kg}\) and \(v = 5.00 \mathrm{m} / \mathrm{s}\), we can calculate the initial total kinetic energy.
02

Calculate the gravitational potential energy at the highest point

When the sphere reaches its highest point on the incline, it comes to a stop; hence all its initial kinetic energy is transformed into gravitational potential energy. We can represent this as: PE_gravity = \(mgh\) Where \(h\) is the vertical height the sphere rises above the horizontal surface.
03

Use the conservation of mechanical energy

The conservation of mechanical energy states that the sum of kinetic and potential energies is constant. In this exercise, we can equate the initial total kinetic energy to the gravitational potential energy at the highest point to find the vertical height \(h\): KE_total = PE_gravity
04

Calculate the vertical height \(h\)

Substitute the values and equations derived from Steps 1-3, and solve for \(h\): \(\frac{1}{2} mv^2 + \frac{1}{2} Iω^2 = mgh\) \(\Rightarrow h = \frac{v^2}{2g} + \frac{Iω^2}{2mg} = \frac{v^2}{2g} + \frac{v^2}{2g} × (\frac{2}{5})\) Substitute the given values of \(m = 0.600 \mathrm{kg}\) and \(v = 5.00 \mathrm{m} / \mathrm{s}\), and use \(g = 9.81 \mathrm{m} / \mathrm{s^2}\) for the acceleration due to gravity. Solve for the vertical height \(h\).
05

Calculate the distance along the incline

Now, knowing the vertical height the sphere rises above the horizontal surface, we can find the distance along the incline by using the relation: \(h= L \sin θ\) Where \(L\) is the distance along the inclined plane and \(θ\) is the angle of inclination. Given, \(θ = 30°\), use the calculated value of \(h\) from Step 4 to calculate \(L\). Thus, following these steps, we can find the vertical height above the horizontal surface to which the sphere rises on the incline.

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Most popular questions from this chapter

A hollow cylinder, a uniform solid sphere, and a uniform solid cylinder all have the same mass \(m .\) The three objects are rolling on a horizontal surface with identical transnational speeds \(v .\) Find their total kinetic energies in terms of \(m\) and \(v\) and order them from smallest to largest.
A Ferris wheel rotates because a motor exerts a torque on the wheel. The radius of the London Eye, a huge observation wheel on the banks of the Thames, is \(67.5 \mathrm{m}\) and its mass is \(1.90 \times 10^{6} \mathrm{kg} .\) The cruising angular speed of the wheel is $3.50 \times 10^{-3} \mathrm{rad} / \mathrm{s} .$ (a) How much work does the motor need to do to bring the stationary wheel up to cruising speed? [Hint: Treat the wheel as a hoop.] (b) What is the torque (assumed constant) the motor needs to provide to the wheel if it takes 20.0 s to reach the cruising angular speed?
A tower outside the Houses of Parliament in London has a famous clock commonly referred to as Big Ben, the name of its 13 -ton chiming bell. The hour hand of each clock face is \(2.7 \mathrm{m}\) long and has a mass of \(60.0 \mathrm{kg}\) Assume the hour hand to be a uniform rod attached at one end. (a) What is the torque on the clock mechanism due to the weight of one of the four hour hands when the clock strikes noon? The axis of rotation is perpendicular to a clock face and through the center of the clock. (b) What is the torque due to the weight of one hour hand about the same axis when the clock tolls 9: 00 A.M.?
A flywheel of mass 182 kg has an effective radius of \(0.62 \mathrm{m}\) (assume the mass is concentrated along a circumference located at the effective radius of the flywheel). (a) How much work is done to bring this wheel from rest to a speed of 120 rpm in a time interval of 30.0 s? (b) What is the applied torque on the flywheel (assumed constant)?
A hollow cylinder, of radius \(R\) and mass \(M,\) rolls without slipping down a loop-the-loop track of radius \(r .\) The cylinder starts from rest at a height \(h\) above the horizontal section of track. What is the minimum value of \(h\) so that the cylinder remains on the track all the way around the loop? A hollow cylinder, of radius \(R\) and mass \(M,\) rolls without slipping down a loop-the- loop track of radius \(r .\) The cylinder starts from rest at a height \(h\) above the horizontal section of track. What is the minimum value of \(h\) so that the cylinder remains on the track all the way around the loop?
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