Q#9
Two uniform solid spheres having unequal masses and unequal radii are released from the rest from the same height on a rough incline. If the spheres roll without slipping,(a) the heavier sphere reaches the bottom first
(b) the bigger sphere reaches the bottom first
(c) the two spheres reach the bottom together
(d) the information given is not sufficient to tell which sphere will reach the bottom first.
Answer: (c)
When the solid sphere rolls without slipping, its linear acceleration is given as,
a = (5/7) gsinθ {where θ is the angl of inclination with the horizontal plane.}
Clearly the linear acceleration is free of mass and the radius. So both the spheres will have equal accelerations and they will reach the bottom together.
Q#10
A hollow sphere and a solid sphere having same mass and same radii are rolled down a rough inclined plane.
(a) the hollow sphere reaches the bottom first.
(b) the solid sphere reaches the bottom with greater speed.
(c) the solid sphere reaches the bottom with greater kinetic energy
(d) the two-sphere will reach the bottom with same linear momentum.
Answer: (b)
For a hollow sphere, the linear acceleration
a = (3/5)gsinΦ = 0.60g.sinθ
For solid sphere
a = (5/7)gsinθ = 0.71gsinθ
Since 'a' for solid sphere is more so it will reach the bottom with a greater speed.
Since both the spheres are released from the same position, their initial potential energy mgh will be the same. So when they reach the bottom they lose same amount of P.E. and this loss of P.E. will be the gain in K.E. So both the spheres will have same K.E.at the bottom. Option (c) is not correct.
Since the solid sphere reaches the bottom with a greater speed and both have same mass, they can not have the same linear momentum. Thus option (d) is not correct.
Q#11
A sphere can not roll on
(a) a smooth horizontal surface
(b) a smooth inclined surface
(c) a rough horizontal surface
(d) a rough inclined surface.
Answer: (b)
On a smooth inclined surface there will be no friction to produce a torque, so no rolling.
Q#12
In rear wheel drive cars, the engine rotates the rear wheels and the front wheels rotate only because the car moves. If such a car accelerates on a horizontal road, the friction
(a) on the rear wheels is in the forward direction.
(b) on the front wheels is in the backward direction.
(c) on the rear wheels has larger magnitude than the friction on the front wheels.
(d) on the car is in the backward direction.
Answer: (a), (b), (c)
When accelerating the torque on the rear wheel tries the part in contact with the road to slide backward so the friction on it is forward. Option (a).
(a) on the rear wheels is in the forward direction.
(b) on the front wheels is in the backward direction.
(c) on the rear wheels has larger magnitude than the friction on the front wheels.
(d) on the car is in the backward direction.
Answer: (a), (b), (c)
When accelerating the torque on the rear wheel tries the part in contact with the road to slide backward so the friction on it is forward. Option (a).
The body of the car tries to push the front wheel in the forward direction, hence the force of friction is in the backward direction. Option (b).
The driving rear wheel has to produce a force in the forward direction (through the torque) to push the whole mass of the car while the torque on the front wheel has just to overcome the inertia of the wheel. Hence the friction on the rear wheel is more than the front wheel. Option (c).
Net friction on the car is in forward direction because rear wheel friction is larger and in forward direction. Hence option (d) is not correct.
Q#13
A sphere can roll on a surface inclined at an angle θ if the friction coefficient is more than (2/7)gsinθ. Suppose the friction coefficient is (1/7)g.sinθ. If a sphere is released from rest on the incline,
(a) it will stay at rest
(b) it will make pure translation motion
(c) it will translate and rotate about the center
(d) the angular momentum of the sphere about its center will remain constant.
Answer: (c)
Since there is some friction which will produce a torque on the sphere and there is no othe torque to balance it. Hnce the sphere will not stay at rest. Option (a) is not correct. Due to the torque produced by the friction it will have some rotational motion also. Hence option (b) is not correct.
(a) it will stay at rest
(b) it will make pure translation motion
(c) it will translate and rotate about the center
(d) the angular momentum of the sphere about its center will remain constant.
Answer: (c)
Since there is some friction which will produce a torque on the sphere and there is no othe torque to balance it. Hnce the sphere will not stay at rest. Option (a) is not correct. Due to the torque produced by the friction it will have some rotational motion also. Hence option (b) is not correct.
From the question the friction coefficient is = (1/7)gsinθ which is lessthan (2/7)gsinθ {required for rolling only} so it will translate and rotate about the center. Option (c) is correct.
As the sphere go down the inclined plane its angular velocity will get increasing. So, the angular momentum of the sphere about its center will continue increasing. Option (d) is not correct.
Q#14
A sphere is rolled on a rough horizontal surface. It gradually slows down and stops. The force of friction tries to
(a) decrease the linear velocity
(b) increase the angular velocity
(c) increase the linear momentum
(d) decrease the angular velocity.
Answer: (a), (b)
Explanation: When a sphere is pushed on a rough horizontal surface to roll, it is given some linear speed and it ties to slip on the surface which is opposed by the force of friction so it tries to decrease yhe linear velocity and due to its direction it tries to increase the angular velocity. Hence option (a) and (b) and not the options (c) and (d).
A sphere is rolled on a rough horizontal surface. It gradually slows down and stops. The force of friction tries to
(a) decrease the linear velocity
(b) increase the angular velocity
(c) increase the linear momentum
(d) decrease the angular velocity.
Answer: (a), (b)
Explanation: When a sphere is pushed on a rough horizontal surface to roll, it is given some linear speed and it ties to slip on the surface which is opposed by the force of friction so it tries to decrease yhe linear velocity and due to its direction it tries to increase the angular velocity. Hence option (a) and (b) and not the options (c) and (d).
Q#15
Figure (10-Q5) shows a smooth inclined plane fixed in a car accelerating on a horizontal road. The angle of incline θ is related to the acceleration of the car as a = gtanθ. If the sphere is set in pure rolling on the incline
(a) it will continue pure rolling
(b) it will slip down the plane
(c) its linear velocity will increase
(d) its linear velocity will decrease.
Answer: (a)
To analize it we have to apply a pseudo force = ma in the opposite direction of a. Its component
Figure (10-Q5) shows a smooth inclined plane fixed in a car accelerating on a horizontal road. The angle of incline θ is related to the acceleration of the car as a = gtanθ. If the sphere is set in pure rolling on the incline
(a) it will continue pure rolling
(b) it will slip down the plane
(c) its linear velocity will increase
(d) its linear velocity will decrease.
Answer: (a)
To analize it we have to apply a pseudo force = ma in the opposite direction of a. Its component
Component of weight along the plane = mgsinθ {towards down the plane}
So net force on the sphere along the plane = mgsinθ – mgsinθ = 0
The perpendicular component of the weight and pseudo force is balanced by the normal force. So it will continue pure rolling.
So net force on the sphere along the plane = mgsinθ – mgsinθ = 0
The perpendicular component of the weight and pseudo force is balanced by the normal force. So it will continue pure rolling.
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