[1] A lawn roller, which is a uniform cylinder of mass M and radius R, is being pulled on a horizontal lawn with a force F that passes through its center of mass, at an angle above the horizontal. As a result, the lawn roller rolls without slipping. Find (a) The angular acceleration a of the roller and the linear acceleration of its center of mass, (b) the force of friction on the roller. (c) If the coefficient of static friction between the roller and the lawn is , what is the maximum value of the force F that can be applied to the roller without causing it to slip?
[1] A lawn roller, which is a uniform cylinder of mass M and radius R, is being pulled on a horizontal lawn with a force F that passes through its center of mass, at an angle above the horizontal. As a result, the lawn roller rolls without slipping. Find (a) The angular acceleration a of the roller and the linear acceleration of its center of mass, (b) the force of friction on the roller. (c) If the coefficient of static friction between the roller and the lawn is , what is the maximum value of the force F that can be applied to the roller without causing it to slip?
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![[1] A lawn roller, which is a uniform cylinder of mass M and radius R, is being pulled on a
horizontal lawn with a force F that passes through its center of mass, at an angle above
the horizontal. As a result, the lawn roller rolls without slipping. Find
(a) The angular acceleration a of the roller and the linear acceleration am of its center of
mass,
(b) the force of friction on the roller.
(c) If the coefficient of static friction between the roller and the lawn is , what is the
maximum value of the force F that can be applied to the roller without causing it to
slip?
[2] A uniform, rigid hoop of radius R and mass M is pivoted at a point A, on its perimeter. It
is able to rotate frictionlessly about point A in the vertical plane in which it lies. A small
piece of clay of the same mass M is shot towards its center at a horizontal speed v
sticking immediately to point B on the hoop. Point B is at the same horizontal level as O,
the center of the hoop. As a result, the hoop (with the clay on it) turns around point A for
many revolutions.
(a) For the collision process between the clay and the hoop, is the mechanical energy of
the clay-hoop system conserved? What about the linear momentum and angular
momentum? You must give brief justifications.
(b) What is the angular speed of the hoop, immediately after the clay got stuck at point
B?
(c) What is the minimum possible value of v? Take R = 25 cm for part (c).
[3] A satellite that weighs 1,600 N on the surface of the Earth (where g -9.79 m/s²) is in a
circular SYNCOM (geosynchronous) orbit above Mars (with 0.107 times the mass of the
Earth and 0.529 times the radius of the Earth, which is 6400 km), with a period one
Martian day (24.6 hours). Find
(a) the acceleration of gravity at the Martian surface,
(b) the weight of the satellite on Martian surface,
(c) the altitude of the Mar's SYNCOM orbit,
(d) the speed of the satellite at Mar's SYNCOM orbit.
[4] An open-top, cylindrical container of cross-sectional radius R = 90 cm is filled up with
water is on a table 1.2 m above the ground. When an opening of diameter of 1.0 cm has
been punched at the bottom of the container, water flows out of it and hits the ground 1.6
m away from the wall of the container.
(a) What is the speed of the water shooting out of the opening at this moment?
(b) What is the initial height of the water in the container?
(c) How long does it take to drain the water completely?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdbdf4c8-5459-45de-99f8-cd8d71031beb%2F0ce72a45-9905-4cb0-9bde-f6cef006e656%2Fq4e4i4y_processed.png&w=3840&q=75)
Transcribed Image Text:[1] A lawn roller, which is a uniform cylinder of mass M and radius R, is being pulled on a
horizontal lawn with a force F that passes through its center of mass, at an angle above
the horizontal. As a result, the lawn roller rolls without slipping. Find
(a) The angular acceleration a of the roller and the linear acceleration am of its center of
mass,
(b) the force of friction on the roller.
(c) If the coefficient of static friction between the roller and the lawn is , what is the
maximum value of the force F that can be applied to the roller without causing it to
slip?
[2] A uniform, rigid hoop of radius R and mass M is pivoted at a point A, on its perimeter. It
is able to rotate frictionlessly about point A in the vertical plane in which it lies. A small
piece of clay of the same mass M is shot towards its center at a horizontal speed v
sticking immediately to point B on the hoop. Point B is at the same horizontal level as O,
the center of the hoop. As a result, the hoop (with the clay on it) turns around point A for
many revolutions.
(a) For the collision process between the clay and the hoop, is the mechanical energy of
the clay-hoop system conserved? What about the linear momentum and angular
momentum? You must give brief justifications.
(b) What is the angular speed of the hoop, immediately after the clay got stuck at point
B?
(c) What is the minimum possible value of v? Take R = 25 cm for part (c).
[3] A satellite that weighs 1,600 N on the surface of the Earth (where g -9.79 m/s²) is in a
circular SYNCOM (geosynchronous) orbit above Mars (with 0.107 times the mass of the
Earth and 0.529 times the radius of the Earth, which is 6400 km), with a period one
Martian day (24.6 hours). Find
(a) the acceleration of gravity at the Martian surface,
(b) the weight of the satellite on Martian surface,
(c) the altitude of the Mar's SYNCOM orbit,
(d) the speed of the satellite at Mar's SYNCOM orbit.
[4] An open-top, cylindrical container of cross-sectional radius R = 90 cm is filled up with
water is on a table 1.2 m above the ground. When an opening of diameter of 1.0 cm has
been punched at the bottom of the container, water flows out of it and hits the ground 1.6
m away from the wall of the container.
(a) What is the speed of the water shooting out of the opening at this moment?
(b) What is the initial height of the water in the container?
(c) How long does it take to drain the water completely?
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