Solution For Problem # 1No. If an object is heavier the force of gravity is greater, but since it has greater mass the acceleration is the same, so it moves at the same speed (if we neglect air resistance). If we look at Newton's second law, F = ma. The force of gravity is F = mg, where m is the mass of the object and g is the acceleration due to gravity.
Equating, we have mg = ma. Therefore, a = g.
If there was no air resistance, a feather would fall at the same speed as an apple.
Solution For Problem # 2
If an object were completely immersed in a liquid denser than it, the resulting buoyant force would exceed the weight of the object. This is because the weight of the liquid displaced by the object is greater than the weight of the object (since the liquid is denser). As a result, the object cannot remain completely submerged and it floats. The scientific name for this phenomenon is Archimedes Principle.
Solution For Problem # 3
Without loss of generality, we only need to look at the equation for the x-position, since we know that centripetal acceleration points towards the center of the circle. Thus, when θ = 0, the second derivative of x with respect to time must be the centripetal acceleration.
The first derivative of x with respect to time t is:
x = —Rsinθ(dθ/dt)
The second derivative of x with respect to time t is:
x = —Rcosθ(dθ/dt)2—Rsinθ(d2θ/dt2)
In both of the above equations the chain rule of Calculus is used and by assumption θ is a function of time. Therefore, θ can be differentiated with respect to time.
Now, evaluate the second derivative at θ = 0.
We have,
x = —R(dθ/dt)2
The term dθ/dt is usually called the angular velocity, which is the rate of change of the angle θ. It has units of radians/second.
For convenience we can set w ≡ dθ/dt.
Therefore,
x = —Rw2
This is the well-known form for the centripetal acceleration equation.
Solution For Problem # 4
The reason you feel weightless is because there is no force pushing against you, since you are not in contact with anything. Gravity is pulling equally on all the particles in your body. This creates a sensation where no forces are acting on you and you feel weightless. It would be the same sensation as if you were floating in space.
Solution For Problem # 5
Centripetal acceleration is the acceleration an object experiences as it travels a certain velocity along an arc. The centripetal acceleration points towards the center of the arc.
Centrifugal force is the imaginary force an unrestrained object experiences as it moves around an arc. This force acts opposite to the direction of centripetal acceleration. For example, if a car makes a sharp right turn the passengers would tend to slide in their seats away from the center of the turn, towards the left (if they are not wearing their seat belts, that is). The passengers would feel as if they are experiencing a force. This is defined as centrifugal force.
Solution For Problem # 6
Power is the rate of energy being generated or consumed. For example, if an engine produces 1000 Watts of power (where Watts is Joules/second), then after an hour the total energy produced by the engine is 1000 Joules/second X 3600 seconds = 3,600,000 Joules.
Solution For Problem # 7
The answer is (a).
Since the collision is head on and each car is identical and traveling at the same speed, the force of impact experienced by each car is equal and opposite. This means that the impact is the same as hitting a solid wall at 100 km/h.
Solution For Problem # 8
When you swing a hammer you increase its kinetic energy, so that by the time it strikes the nail it imparts a large force which drives the nail into the wood.
The hammer is basically an energy reservoir to which you are adding energy during the course of the swing, and which is released all at once upon impact. This results in the impact force greatly exceeding the maximum force you can exert by just pushing on the nail.
Solution For Problem # 9
This can be solved using an energy method.
We can solve this by equating the potential energy of the bow to the kinetic energy of the arrow.
The bow can be treated as a type of spring. The potential energy of a spring is:
(1/2)kx2, where k is the stiffness and x is the amount the spring is stretched, or compressed.
Therefore, the potential energy PE of the bow is:
PE = (1/2)(200)(0.75)2 = 56.25 J
The kinetic energy of a particle is:
(1/2)mv2, where m is the mass and v is the velocity.
The arrow can be treated as a particle since it is not rotating upon release.
Therefore, the kinetic energy KE of the arrow is:
KE = (1/2)(0.05)v2
If we assume energy is conserved, then
PE = KE
Solving for the velocity of the arrow v we get
v = 47.4 m/s
Solution For Problem # 10
Static friction is greater than kinetic friction.
Static friction exists if the wheels keep rolling on the ice without locking up, resulting in maximum braking force. However, if the wheels lock up then kinetic friction takes over since there is relative slipping between wheel and ice. This reduces the braking force and the car takes longer to stop.
Anti-lock braking systems (ABS) on a vehicle prevent the wheels from locking up when the brakes are applied, thus minimizing the amount of time it takes for the vehicle to reach a complete stop. Also, by preventing the wheels from locking up you have greater control of the vehicle.
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