Q: Two children push on opposite sides of a door during play. One pushes with a force of 175 N at a distance of 0.70 m from the hinges, and the other pushes at a distance of 0.40 m. What force (in N) must the second child exert to keep the door from moving? Assume friction is negligible.

$$F_2$$ = \frac{$$F_1$$ $$r_1$$}{$$r_2$$} = \frac{175 \times 0.70}{0.40} = 306.25\ \text{N}$$

Q: An object with mass $$m = 1\ \text{kg} is $$attached to a spring with a force constant $$k = 40\ \text{N/m}$$ and released from rest when the spring is stretched $$4\ \text{m}. $$What is the magnitude and direction of the force exerted by the spring at this point?

$$F = -kx = -40 \times 4 = -160\ \text{N}$$, 160 N toward the anchor/equilibrium

Q: A child drives a bumper car head-on into the side rail, which exerts a force of $$2700\ \text{N} on $$the car for $$0.400\ \text{s}. $$What impulse is imparted by this force?

$$J = F \Delta t = 2700 \times 0.400 = 1080\ \text{N} \cdot \text{s}$$

Q: A 2650-kg cannon is mounted so that it can recoil only in the horizontal direction. Calculate its recoil velocity if $$m = 16.0\ \text{kg}$$ shell is fired at $$480\ \text{m/s} at an $$angle of $$20.0^\circ$$ above the horizontal.

$$v_{\text{recoil}} = -2.10\ \text{m/s}$$

Q: Suppose you press a $$5\ \text{kg}$$ mass into a spring with stiffness $$k = 200\ \text{N/m}$$, compressing it by $$20\ \text{cm}$$ from equilibrium. What is the maximum potential energy stored in the spring?

$$U = \frac{1}{2} k x^2$ = \frac{1}{2} \times 200 \times (0.20)^2$ = 4\ \text{J}$$

Q: A tennis ball with mass $$0.058\ \text{kg} is $$thrown at a door to just barely make it close, overcoming a frictional torque of $$0.5\ \text{N} \cdot \text{m}. If $$the ball is thrown at the edge of the door $$0.9\ \text{m}$$ from the hinge, what minimum speed must the ball have?

$$v = \frac{\tau}{m r} = \frac{0.5}{0.058 \times 0.9} \approx 9.6\ \text{m/s}$$

Q: A block of mass $$m$$ slides across a nearly frictionless surface until it loses contact with a spring. Some time later, the mass encounters a concrete floor and continues to slide. The coefficient of kinetic friction between the mass and the concrete is $$0.15. If $$the block's initial speed is $$2.0\ \text{m/s}$$, how far does the mass slide before coming to rest?

$$d = \frac{v^2$}{2 \mu g} = \frac{(2.0)^2$}{2 \times 0.15 \times 9.8} \approx 1.36\ \text{m}$$

Q: A spring with force constant $$k = 40\ \text{N/m} is $$stretched $$2\ \text{m}$$ from equilibrium. What force is exerted on the object now?

$$F = -kx = -40 \times 2 = -80\ \text{N}$$

Q: A $$5\ \text{kg}$$ mass is pressed into a spring with stiffness $$k = 200\ \text{N/m}$$ and compressed by $$20\ \text{cm}. $$What is the speed of the mass right after it loses contact with the spring?

$$v = \sqrt{\frac{k x^2$}{m}} = \sqrt{\frac{200 \times (0.20)^2$}{5}} \approx 1.26\ \text{m/s}$$

Question 1

Two children push on opposite sides of a door during play. One pushes with a force of 175 N at a distance of 0.70 m from the hinges, and the other pushes at a distance of 0.40 m. What force (in N) must the second child exert to keep the door from moving? Assume friction is negligible.

Accepted Answer

$$F_2$$ = \frac{$$F_1$$ $$r_1$$}{$$r_2$$} = \frac{175 	imes 0.70}{0.40} = 306.25\ 	ext{N}$$

Question 2

An object with mass $$m = 1\ 	ext{kg} is $$attached to a spring with a force constant $$k = 40\ 	ext{N/m}$$ and released from rest when the spring is stretched $$4\ 	ext{m}. $$What is the magnitude and direction of the force exerted by the spring at this point?

Accepted Answer

$$F = -kx = -40 	imes 4 = -160\ 	ext{N}$$, 160 N toward the anchor/equilibrium

Question 3

A uniform beam of length $$2.0\ 	ext{m} is $$supported by a cable at one end and the force of friction at the other. The cable makes an angle of $$30^\circ$$ with the horizontal. If the coefficient of static friction between the wall and the beam is $$\mu_s = 0.560$$, what is the minimum distance $$x$$ from point A at which an additional weight $$2w ($$twice the weight of the rod) can be hung without causing the rod to slip at point A?

Accepted Answer

$$x = 1.12\ 	ext{m}$$

Question 4

A child drives a bumper car head-on into the side rail, which exerts a force of $$2700\ 	ext{N} on $$the car for $$0.400\ 	ext{s}. $$What impulse is imparted by this force?

Accepted Answer

$$J = F \Delta t = 2700 	imes 0.400 = 1080\ 	ext{N} \cdot 	ext{s}$$

Question 5

A 2650-kg cannon is mounted so that it can recoil only in the horizontal direction. Calculate its recoil velocity if $$m = 16.0\ 	ext{kg}$$ shell is fired at $$480\ 	ext{m/s} at an $$angle of $$20.0^\circ$$ above the horizontal.

Accepted Answer

$$v_{	ext{recoil}} = -2.10\ 	ext{m/s}$$

Question 6

A $$1000\ 	ext{kg}$$ car is moving eastward at $$20\ 	ext{m/s}. It $$collides with a $$1500\ 	ext{kg}$$ van traveling northward at $$30\ 	ext{m/s}. If $$the wreckage of the car and truck moves together after the collision, what type of collision is this?

Accepted Answer

Perfectly inelastic

Question 7

A ballistic pendulum consists of a $$2.4\ 	ext{kg}$$ block shot by a $$50\ 	ext{g}$$ bullet. The bullet impacts the block at $$120\ 	ext{m/s}$$ and embeds itself in the block. What quantity is conserved in the collision?

Accepted Answer

Linear momentum

Question 8

Suppose you press a $$5\ 	ext{kg}$$ mass into a spring with stiffness $$k = 200\ 	ext{N/m}$$, compressing it by $$20\ 	ext{cm}$$ from equilibrium. What is the maximum potential energy stored in the spring?

Accepted Answer

$$U = \frac{1}{2} k x^2$ = \frac{1}{2} 	imes 200 	imes (0.20)^2$ = 4\ 	ext{J}$$

Question 9

A tennis ball with mass $$0.058\ 	ext{kg} is $$thrown at a door to just barely make it close, overcoming a frictional torque of $$0.5\ 	ext{N} \cdot 	ext{m}. If $$the ball is thrown at the edge of the door $$0.9\ 	ext{m}$$ from the hinge, what minimum speed must the ball have?

Accepted Answer

$$v = \frac{	au}{m r} = \frac{0.5}{0.058 	imes 0.9} \approx 9.6\ 	ext{m/s}$$

Question 10

A block of mass $$m$$ slides across a nearly frictionless surface until it loses contact with a spring. Some time later, the mass encounters a concrete floor and continues to slide. The coefficient of kinetic friction between the mass and the concrete is $$0.15. If $$the block's initial speed is $$2.0\ 	ext{m/s}$$, how far does the mass slide before coming to rest?

Accepted Answer

$$d = \frac{v^2$}{2 \mu g} = \frac{(2.0)^2$}{2 	imes 0.15 	imes 9.8} \approx 1.36\ 	ext{m}$$

Question 11

A spring with force constant $$k = 40\ 	ext{N/m} is $$stretched $$2\ 	ext{m}$$ from equilibrium. What force is exerted on the object now?

Accepted Answer

$$F = -kx = -40 	imes 2 = -80\ 	ext{N}$$

Question 12

A $$5\ 	ext{kg}$$ mass is pressed into a spring with stiffness $$k = 200\ 	ext{N/m}$$ and compressed by $$20\ 	ext{cm}. $$What is the speed of the mass right after it loses contact with the spring?

Accepted Answer

$$v = \sqrt{\frac{k x^2$}{m}} = \sqrt{\frac{200 	imes (0.20)^2$}{5}} \approx 1.26\ 	ext{m/s}$$

Question 13

A block of mass $$m is $$attached to a spring and oscillates with amplitude $$A. At $$what point does the force vanish?

Accepted Answer

At the equilibrium (natural) length of the spring: $$x = 0$$

Question 14

A $$2.4\ 	ext{kg}$$ block is shot by a $$50\ 	ext{g}$$ bullet at $$120\ 	ext{m/s}$$ and swings up to a certain height. What is the maximum vertical displacement of the block?

Accepted Answer

$$h = \frac{(m_b v_b)^2$}{2 (m_b + m_{block})^2$ g}$$

Question 15

A $$1000\ 	ext{kg}$$ car moving east at $$20\ 	ext{m/s}$$ collides with a $$1500\ 	ext{kg}$$ van moving north at $$30\ 	ext{m/s}. $$What is the magnitude of the velocity of the wreckage immediately after the collision?

Accepted Answer

$$v = \frac{\sqrt{(1000 	imes 20)^2$ + (1500 	imes 30)^2$}}{2500} \approx 22.8\ 	ext{m/s}$$

Question 16

A $$200\ 	ext{kg}$$ bumper car initially moving at $$3.30\ 	ext{m/s}$$ collides with a stationary $$200\ 	ext{kg}$$ car. If the collision is perfectly inelastic, what is the final velocity of the combined cars?

Accepted Answer

$$v_f = \frac{m_1$ v_1$ + m_2$ v_2$}{m_1$ + m_2$} = \frac{200 	imes 3.30 + 200 	imes 0}{400} = 1.65\ 	ext{m/s}$$

Forces, Equilibrium, Momentum, and Collisions: Study Notes

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