Concept explainers
(a)
The gravitational potential energy of the graph-Earth system at the grape’s initial position.
(a)
Answer to Problem 82PQ
The gravitational potential energy of the graph-Earth system at the grape’s initial position is
Explanation of Solution
The grapes position at different heights is shown below.
Write the expression for the radius of bowl.
Here,
Write the expression for the gravitational potential energy.
Here,
Conclusion:
Initially grape is resting at upper edge. Thus, initial height is equal to radius of bowl.
Substitute
Substitute
Here,
Therefore, the gravitational potential energy of the graph-Earth system at the grape’s initial position is
(b)
The kinetic energy of the grape when it reaches the bottom of the bowl.
(b)
Answer to Problem 82PQ
The kinetic energy of the grape when it reaches the bottom of the bowl is
Explanation of Solution
Take the bottom of the bowl as
Write conservation of energy equation as the grape moves from top of the bowl to bottom of bowl.
Here,
Conclusion:
In problem it is given that initially the grape is at rest at upper edge of bowl. At bottom of bowl potential energy is zero, since
Substitute
Substitute
Therefore, the kinetic energy of the grape when it reaches the bottom of the bowl is
(c)
The speed of the grape when it reaches the bottom of the bowl.
(c)
Answer to Problem 82PQ
The speed of the grape when it reaches the bottom of the bowl is
Explanation of Solution
Kinetic energy of the grape at bottom of bowl is obtained as
Write the expression for kinetic energy of grape.
Here,
Rearrange above equation to get
Conclusion:
Substitute
Therefore, the speed of the grape when it reaches the bottom of the bowl is
(d)
The potential and kinetic energies of the grape when it reaches a point that is height
(d)
Answer to Problem 82PQ
The potential energy of the grape when it reaches a point that is height
Explanation of Solution
Rewrite equation (I) to get potential energy at a height.
Write conservation of energy equation as the grape moves from the top of the bowl to a height
Here,
Conclusion:
Substitute
In problem it is given that initially the ball is at rest at upper edge of bowl. Kinetic energy is zero at top edge of bowl. At top edge of bowl potential energy is
Substitute
Therefore, the potential energy of the grape when it reaches a point that is height
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Chapter 8 Solutions
Physics for Scientists and Engineers: Foundations and Connections
- A small block of mass m = 200 g is released from rest at point along the horizontal diameter on the inside of a frictionless, hemispherical bowl of radius R = 30.0 cm (Fig. P7.45). Calculate (a) the gravitational potential energy of the block-Earth system when the block is at point relative to point . (b) the kinetic energy of the block at point , (c) its speed at point , and (d) its kinetic energy and the potential energy when the block is at point . Figure P7.45 Problems 45 and 46.arrow_forwardA block is hung from a vertical spring. The spring stretches (h = 0.0650 m) as shown for a particular instant in time in Figure P8.26. Consider the Earth, spring, and block to be in the system. If m = 0.865 kg and k = 125 N/m, find the change in the systems potential energy between the two times depicted in the figure. FIGURE P8.26arrow_forwardThe Earths perihelion distance (closest approach to the Sun) is rp = 1.48 1011 m, and its aphelion distance (farthest point) is rA = 1.52 1011 m. What is the change in the SunEarths gravitational potential energy as the Earth moves from aphelion to perihelion? What is the change in its gravitational potential energy from perihelion to aphelion?arrow_forward
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