Question 1

A block and ball system is released from rest at the top of a circular trajectory. To ensure the string remains taut at the top, what is the minimum velocity $$v_{min}$$ required at the top, given the combined mass is $$(m_1$ + m_2$)$$ and the radius is $$L$$?

Accepted Answer

$$v_{min} = \sqrt{gL}$$

Question 2

In Problem 1b, when the block-ball system reaches the bottom of the trajectory, all potential energy is converted to kinetic energy. If the system starts at height $$2L$$, what is the expression for the final speed $$v_{fin} at $$the bottom?

Accepted Answer

$$v_{fin} = \sqrt{5gL}$$

Question 3

After a collision in Problem 1c, the block-ball system moves with a new velocity. If $$m_1$$ and $$m_2$$ are the masses, and the block must move at $$\sqrt{gL}$$, what is the velocity $$v_1 of$$ $$m_1$$ after the collision?

Accepted Answer

$$v_1$$ = \frac{$$m_1$$ + $$m_2$$}{$$m_1$$} \sqrt{gL}$$

Question 4

The escape velocity from the surface of the Earth is derived using gravitational potential energy and kinetic energy. Which formula correctly gives the escape velocity $$v_{esc} at $$distance $$r$$ from the center of the Earth?

Accepted Answer

$$v_{esc} = \sqrt{\frac{2GM}{r}}$$

Question 5

A satellite is launched with velocity $$v_{esc}$$ from a distance $$d = 11R/10$$ from Earth's center. What is the value of $$v_{esc} at $$this distance?

Accepted Answer

$$v_{esc} = \sqrt{\frac{20GM}{11R}}$$

Question 6

Immediately after launch, the energy of the satellite is given by $$\frac{1}{2}mv^2$ - \frac{GMm}{R}. If$$ $$v = v_{esc}$$ and $$d = 11R/10$$, what is the total energy right after launch?

Accepted Answer

$$-\frac{5GMm}{22R}$$

Question 7

According to the work-energy theorem, the work required to move a satellite from $$d = 11R/10 to a $$stable circular orbit at $$R is $$equal to the change in kinetic energy. What is the required work $$W$$?

Accepted Answer

$$W = \frac{3GMm}{11R}$$

Question 8

Rain falls at a constant mass rate $$\beta$$ onto a boat of mass $$m_0. $$What is the expression for the total mass $$m(t) of $$the boat after time $$t$$?

Accepted Answer

$$m(t) = m_0$ + \beta t$$

Question 9

The drag force $$F_D$$ acting on the boat is proportional to its velocity $$v$$ and is given by $$F_D = -bv^2$. If the boat's mass increases due to rain, what is the differential equation describing the velocity v$?

Accepted Answer

$$\frac{dv}{dt} = -\frac{b}{m(t)}v^2$

Question 10

In Problem 3e, integrating the velocity equation for the boat yields $$v(t) = \frac{v_0$}{\beta + v_0$ \log\left(\frac{m_0$ + \beta t}{m_0$}ight)}. $$What happens to $$v(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$v(t) 	o 0$$

Question 11

A block falls with acceleration $$a$$ while attached to a rotating pulley of mass $$M$$ and radius $$R. $$What is the moment of inertia $$I_{pulley}$$ for the pulley?

Accepted Answer

$$I_{pulley} = \frac{1}{2}MR^2$

Question 12

In Problem 4, the tension $$T_1 in $$the string is given by $$T_1 = mg$$ - \frac{2mh}{t_$$f^2$$}$$. What does the term $$\frac{2mh}{t_$$f^2$$}$$ represent?

Accepted Answer

The net force required for the block's acceleration

Question 13

If the block falls a distance $$h in $$time $$t_f$$, what is the acceleration $$a of $$the block?

Accepted Answer

$$a = \frac{2h}{t_f^2$}$$

Question 14

In the context of Problem 2e, why is mechanical energy not conserved after the satellite is launched but before it is in a stable orbit?

Accepted Answer

Because work must be done to change the satellite's kinetic energy to achieve a stable orbit

Question 15

In Problem 3, the buoyant force on the boat is counteracted by the weight of the boat and the momentum carried by the rain. What would a scale under the boat read at any instant?

Accepted Answer

$$m_0$$ + \beta t$$

Question 16

In Problem 4c, the rotational acceleration $$\alpha of $$the pulley is related to the block's acceleration $$a by$$ $$\alpha = \frac{a}{R}. $$What physical principle does this relationship illustrate?

Accepted Answer

The kinematic constraint between linear and angular acceleration

Question 17

In Problem 4d, the total moment of inertia $$I_{tot}$$ for the system is $$I_{tot} = I + I_0$. If I$ is the moment of inertia of the spool and I_0$ is that of the object, what is the effect of increasing I_0$ on the system's angular acceleration?

Accepted Answer

It decreases the angular acceleration

Physics 7A: Classical Mechanics - Problem Set Solutions and Concepts

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