Q: If the car's speed at the top of the hill exceeds a critical value $$v_{\text{crit}}$$, the car loses contact with the ground. What is the value of $$v_{\text{crit}} in $$terms of $$g$$ and $$r$$?

$$v_{\text{crit}} = \sqrt{gr}$$

Q: A car is moving in a circle of radius $$R on a $$frictionless banked road at speed $$v. $$What is the angle $$\theta$$ the road must be banked so that no friction is required?

$$\theta = \tan$$^{-1}$$\left(\frac{v^2$}{gR}\right)$$

Q: Which of the following is the correct vector equation relating the force $$\vec{F}_{\text{car on truck}}$$ exerted by the car on the truck and the force $$\vec{F}_{\text{truck on car}}$$ exerted by the truck on the car?

$$\vec{F}_{\text{car on truck}} = -\vec{F}_{\text{truck on car}}$$

Question 1

A car of mass $$m$$ travels over the top of a hill of radius $$r at $$speed $$v. $$What is the normal force $$N$$ exerted by the road on the car at the top of the hill?

Accepted Answer

$$N = mg - \frac{mv^2$}{r}$$

Question 2

If the car's speed at the top of the hill exceeds a critical value $$v_{	ext{crit}}$$, the car loses contact with the ground. What is the value of $$v_{	ext{crit}} in $$terms of $$g$$ and $$r$$?

Accepted Answer

$$v_{	ext{crit}} = \sqrt{gr}$$

Question 3

A bullet of mass $$m_1 = 50$ g is embedded in a block of mass m_2$ = 950 g. If $$the bullet is fired at $$15 m/s $$and the block is initially at rest, what is the speed of the block and embedded bullet immediately after the collision?

Accepted Answer

$$v = \frac{m_1$ v_1$}{m_1$ + m_2$}$$

Question 4

You and a friend each pull on opposite ends of a rope with a force of $$1000 N. $$The rope breaks if the tension exceeds $$1500 N. $$What happens?

Accepted Answer

The rope does not break because the tension is $$1000 N.$$

Question 5

A $$1000 kg $$car pushes a $$2000 kg $$truck with a dead battery. The car exerts a horizontal force of $$4000 N on $$the truck. What is the magnitude of the force exerted by the truck on the car?

Accepted Answer

$$4000 N$$, in the opposite direction

Question 6

A car is moving in a circle of radius $$R on a $$frictionless banked road at speed $$v. $$What is the angle $$	heta$$ the road must be banked so that no friction is required?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{v^2$}{gR}ight)$$

Question 7

A particle of mass $$2 kg $$moves along the $$x-$$axis with potential energy $$U(x) as $$shown. At $$x = 2 m$$, the slope of $$U(x) is $$negative. What is the direction of the force $$F_x on $$the particle?

Accepted Answer

Positive $$x$$ direction

Question 8

Given a potential energy curve $$U(x)$$ with a local minimum at $$x_0$$, what type of equilibrium does $$x_0$$ represent?

Accepted Answer

Stable equilibrium

Question 9

A particle starts at $$x = 3 m $$with initial speed $$v_0$$ and moves to the right. What is the minimum value of $$v_0$$ required for the particle to reach $$x = 0 m$$, given the potential energy curve $$U(x)$$?

Accepted Answer

$$v_0$$ = \sqrt{\frac{2}{m}(U(0) - U(3))}$$

Question 10

Two stars of masses $$M_1$$ and $$M_2$$ are released from rest at a distance $$d$$ apart in deep space. What is the speed of each star just before impact, assuming only gravitational attraction?

Accepted Answer

$$v_1$$ = \sqrt{\frac{2G($$M_1$$ + $$M_2$$)}{d}\frac{$$M_2$$}{$$M_1$$ + $$M_2$$}}$$

Question 11

An elevator descends and slows to a stop just before reaching the first floor. Which of the following best describes the direction of the net force on the elevator during this slowing phase?

Accepted Answer

Upward

Question 12

If you tie a rope into a loop and pull on both ends in the same direction, what is the maximum tension in the rope?

Accepted Answer

The tension is equal to the force applied by one person

Question 13

Which of the following is the correct vector equation relating the force $$\vec{F}_{	ext{car on truck}}$$ exerted by the car on the truck and the force $$\vec{F}_{	ext{truck on car}}$$ exerted by the truck on the car?

Accepted Answer

$$\vec{F}_{	ext{car on truck}} = -\vec{F}_{	ext{truck on car}}$$

Question 14

A car moves in a circle of radius $$R at $$constant speed $$v. $$What is the direction of the net force acting on the car?

Accepted Answer

Toward the center of the circle

Question 15

A block of mass $$m$$ slides on a frictionless surface and is attached to a spring with spring constant $$k. $$What is the period $$T of $$oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{m}{k}}$$

Question 16

A car of mass $$m is at $$rest at the top of a frictionless hill of height $$h. $$What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{2gh}$$

Question 17

A $$2 kg $$particle moves in a potential $$U(x)$$ with a local maximum at $$x = 1 m. $$What type of equilibrium is this?

Accepted Answer

Unstable equilibrium

Question 18

A car of mass $$m$$ moves in a circle of radius $$r at $$speed $$v. $$What is the magnitude of the centripetal force required?

Accepted Answer

$$F = \frac{mv^2$}{r}$$

Question 19

A block of mass $$m is $$dropped from height $$h$$ and rebounds to height $$h/2. $$What fraction of its initial energy is lost?

Accepted Answer

$$\frac{1}{2}$$

Physics 407: Practice Final Exam Study Notes

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