chap. 13 kinetics of particles: force and acceleration
TRANSCRIPT
Chap. 2 Force Vectors 13-2
APPLICATIONS
The motion of an object depends on the forces acting on it.
Knowing the drag force, how can we determine the acceleration or velocity of the parachutist at any point in time?
A parachutist relies on the atmospheric drag resistance force to limit his velocity.
13-3
APPLICATIONS
A freight elevator is lifted using a motor attached to a cable and pulley system as shown.
How can we determine the tension force in the cable required to lift the elevator at a given acceleration?
Is the tension force in the cable greater than the weight of the elevator and its load?
13-4
13-8
p. 122, 13-10 The crate has a mass of 80 kg and is being towed by a chain which is always directed at 20°from the horizontal as shown. If the magnitude of P is increased until the crate begins to slide, determine the crate’s initial acceleration if the coefficient of static friction is μS = 0.5 and the coefficient of kinetic friction is μK = 0.3.
13-9
13-10
p. 128, 13-36 Blocks A and B each have a mass m. Determine the largest horizontal force P which can be applied to B so that A will not move relative to B. All surfaces are smooth.
13-11
13-12
13-13
p. 138, 13-48 The 2-kg block B and 15-kg cylinder A areconnected to a light cord that passes through a hole in the center of the smooth table. If the block is given a speed of v = 10 m/s ,determine the radius r of the circular path along which it travels.
13-14
13-15
p. 140, 13-61 If the ball has a mass of 30kg and a speed at the instant it is at its lowest point, ,determine the tension in the cord at this instant .Also, determine the angle to which the balls wings and momentarily stops. Neglect the size of the ball.
13-16
13-17
p. 141, 13-72 The 0.8-Mg car travels over the hill having the shape of a parabola. If the driver maintains a constant speed of 9 m/s, determine both the resultant normal force and the resultant frictional force that all the wheels of the car exert on the road at the instant it reaches point A. Neglect the size of the car.
13-18
)2(
)90sin(
)2(
TANGENTIAL AND NORMAL FORCES
If a force P causes the particle to move along a path defined by r = f (θ ), the normal force N exerted by the path on the particle is always perpendicular to the path’s tangent. The frictional force F always acts along the tangent in the opposite direction of motion. The directions of N and F can be specified relative to the radial coordinate by using angle ψ .
13-22
The angle ψ,
defined as the angle between the extended radial line and the tangent to the curve, can be required to solve some problems. It can be determined from the following relationship.
θ θψ
ddr r
dr r d ==tan
If ψ is positive, it is measured counterclockwise from the radial line to the tangent. If it is negative, it is measured clockwise.
Def. PFCF (Central force)C (Center of force) FO ∴ΣMo = 0
∴
.2
areal velocity
dt dAp
dt dA
)(
22
h ab
2.
3. T2~a3 (need correct for text, should be major axis, not minor axis)
13-38
32
32
2
222
2
πππ
π
εθ
εθ
Q
13-39
p. 151, 13-92 If the coefficient of static friction between the conical surface and the block of mass m is , determine the minimum constant angular velocity so that the block does not slide downwards.
13-40
13-41
p. 154, 13-112 The 0.5-kg ball is guided along the vertical circular path r = 2rC cosθ
using the arm OA. If the arm has an angular velocity θ
= 0.4 rad/s and an angular acceleration θ
= 0.8 rad/s2 at the instant θ = 30°,
determine the force of the arm on the bal. Neglect friction and the size of the ball. Set rC = 0.12 m.
13-42
13-43
p. 163, 13-120 The space shuttle is launched with a velocity of 28 000 km/h parallel to the tangent of the earth’s surface at point P and then travels around the elliptical orbit.When it reaches point A, its engines are turned on and its velocity is suddenly increased. Determine the required increase in velocity so that it enters the second elliptical orbit.
13-44
13-45
p. 165, 13-132 The satellite is in an elliptical orbit having an eccentricity of e = 0.5. If its velocity at perigee is vP = 15 Mm/hr , determine its velocity at apogee A and the period of the satellite.
13-46
13-47
2
3
p. 122, 13-10The crate has a mass of 80 kg and is being towed by a chain which is always directed at 20°from the horizontal as shown. If the magnitude of P is increased until the crate begins to slide, determine the crate’s initial acceleration if the coefficient of static friction is S = 0.5 and the coefficient of kinetic friction is K = 0.3.
9
p. 128, 13-36Blocks A and B each have a mass m. Determine the largest horizontal force P which can be applied to B so that A will not move relative to B. All surfaces are smooth.
11
p. 138, 13-48The 2-kg block B and 15-kg cylinder A areconnected to a light cord that passes through a hole in the center of the smooth table. If the block is given a speed of v = 10 m/s ,determine the radius r of the circular path along which it travels.
14
p. 140, 13-61If the ball has a mass of 30kg and a speed at the instant it is at its lowest point, ,determine the tension in the cord at this instant .Also, determine the angle to which the balls wings and momentarily stops. Neglect the size of the ball.
16
p. 141, 13-72The 0.8-Mg car travels over the hill having the shape of a parabola. If the driver maintains a constant speed of 9 m/s, determine both the resultant normal force and the resultant frictional force that all the wheels of the car exert on the road at the instant it reaches point A. Neglect the size of the car.
18
20
21
22
27
28
30
31
32
33
34
35
37
38
p. 151, 13-92If the coefficient of static friction between the conical surface and the block of mass m is , determine the minimum constant angular velocity so that the block does not slide downwards.
40
p. 154, 13-112The 0.5-kg ball is guided along the vertical circular path r = 2rCcos using the arm OA. If the arm has an angular velocity = 0.4 rad/s and an angular acceleration = 0.8 rad/s2 at the instant = 30, determine the force of the arm on the bal. Neglect friction and the size of the ball. Set rC = 0.12 m.
42
p. 163, 13-120The space shuttle is launched with a velocity of 28 000 km/h parallel to the tangent of the earth’s surface at point P and then travels around the elliptical orbit.When it reaches point A, its engines are turned on and its velocity is suddenly increased. Determine the required increase in velocity so that it enters the second elliptical orbit.
44
p. 165, 13-132The satellite is in an elliptical orbit having an eccentricity of e = 0.5. If its velocity at perigee is vP = 15 Mm/hr , determine its velocity at apogee A and the period of the satellite.
46
47
APPLICATIONS
The motion of an object depends on the forces acting on it.
Knowing the drag force, how can we determine the acceleration or velocity of the parachutist at any point in time?
A parachutist relies on the atmospheric drag resistance force to limit his velocity.
13-3
APPLICATIONS
A freight elevator is lifted using a motor attached to a cable and pulley system as shown.
How can we determine the tension force in the cable required to lift the elevator at a given acceleration?
Is the tension force in the cable greater than the weight of the elevator and its load?
13-4
13-8
p. 122, 13-10 The crate has a mass of 80 kg and is being towed by a chain which is always directed at 20°from the horizontal as shown. If the magnitude of P is increased until the crate begins to slide, determine the crate’s initial acceleration if the coefficient of static friction is μS = 0.5 and the coefficient of kinetic friction is μK = 0.3.
13-9
13-10
p. 128, 13-36 Blocks A and B each have a mass m. Determine the largest horizontal force P which can be applied to B so that A will not move relative to B. All surfaces are smooth.
13-11
13-12
13-13
p. 138, 13-48 The 2-kg block B and 15-kg cylinder A areconnected to a light cord that passes through a hole in the center of the smooth table. If the block is given a speed of v = 10 m/s ,determine the radius r of the circular path along which it travels.
13-14
13-15
p. 140, 13-61 If the ball has a mass of 30kg and a speed at the instant it is at its lowest point, ,determine the tension in the cord at this instant .Also, determine the angle to which the balls wings and momentarily stops. Neglect the size of the ball.
13-16
13-17
p. 141, 13-72 The 0.8-Mg car travels over the hill having the shape of a parabola. If the driver maintains a constant speed of 9 m/s, determine both the resultant normal force and the resultant frictional force that all the wheels of the car exert on the road at the instant it reaches point A. Neglect the size of the car.
13-18
)2(
)90sin(
)2(
TANGENTIAL AND NORMAL FORCES
If a force P causes the particle to move along a path defined by r = f (θ ), the normal force N exerted by the path on the particle is always perpendicular to the path’s tangent. The frictional force F always acts along the tangent in the opposite direction of motion. The directions of N and F can be specified relative to the radial coordinate by using angle ψ .
13-22
The angle ψ,
defined as the angle between the extended radial line and the tangent to the curve, can be required to solve some problems. It can be determined from the following relationship.
θ θψ
ddr r
dr r d ==tan
If ψ is positive, it is measured counterclockwise from the radial line to the tangent. If it is negative, it is measured clockwise.
Def. PFCF (Central force)C (Center of force) FO ∴ΣMo = 0
∴
.2
areal velocity
dt dAp
dt dA
)(
22
h ab
2.
3. T2~a3 (need correct for text, should be major axis, not minor axis)
13-38
32
32
2
222
2
πππ
π
εθ
εθ
Q
13-39
p. 151, 13-92 If the coefficient of static friction between the conical surface and the block of mass m is , determine the minimum constant angular velocity so that the block does not slide downwards.
13-40
13-41
p. 154, 13-112 The 0.5-kg ball is guided along the vertical circular path r = 2rC cosθ
using the arm OA. If the arm has an angular velocity θ
= 0.4 rad/s and an angular acceleration θ
= 0.8 rad/s2 at the instant θ = 30°,
determine the force of the arm on the bal. Neglect friction and the size of the ball. Set rC = 0.12 m.
13-42
13-43
p. 163, 13-120 The space shuttle is launched with a velocity of 28 000 km/h parallel to the tangent of the earth’s surface at point P and then travels around the elliptical orbit.When it reaches point A, its engines are turned on and its velocity is suddenly increased. Determine the required increase in velocity so that it enters the second elliptical orbit.
13-44
13-45
p. 165, 13-132 The satellite is in an elliptical orbit having an eccentricity of e = 0.5. If its velocity at perigee is vP = 15 Mm/hr , determine its velocity at apogee A and the period of the satellite.
13-46
13-47
2
3
p. 122, 13-10The crate has a mass of 80 kg and is being towed by a chain which is always directed at 20°from the horizontal as shown. If the magnitude of P is increased until the crate begins to slide, determine the crate’s initial acceleration if the coefficient of static friction is S = 0.5 and the coefficient of kinetic friction is K = 0.3.
9
p. 128, 13-36Blocks A and B each have a mass m. Determine the largest horizontal force P which can be applied to B so that A will not move relative to B. All surfaces are smooth.
11
p. 138, 13-48The 2-kg block B and 15-kg cylinder A areconnected to a light cord that passes through a hole in the center of the smooth table. If the block is given a speed of v = 10 m/s ,determine the radius r of the circular path along which it travels.
14
p. 140, 13-61If the ball has a mass of 30kg and a speed at the instant it is at its lowest point, ,determine the tension in the cord at this instant .Also, determine the angle to which the balls wings and momentarily stops. Neglect the size of the ball.
16
p. 141, 13-72The 0.8-Mg car travels over the hill having the shape of a parabola. If the driver maintains a constant speed of 9 m/s, determine both the resultant normal force and the resultant frictional force that all the wheels of the car exert on the road at the instant it reaches point A. Neglect the size of the car.
18
20
21
22
27
28
30
31
32
33
34
35
37
38
p. 151, 13-92If the coefficient of static friction between the conical surface and the block of mass m is , determine the minimum constant angular velocity so that the block does not slide downwards.
40
p. 154, 13-112The 0.5-kg ball is guided along the vertical circular path r = 2rCcos using the arm OA. If the arm has an angular velocity = 0.4 rad/s and an angular acceleration = 0.8 rad/s2 at the instant = 30, determine the force of the arm on the bal. Neglect friction and the size of the ball. Set rC = 0.12 m.
42
p. 163, 13-120The space shuttle is launched with a velocity of 28 000 km/h parallel to the tangent of the earth’s surface at point P and then travels around the elliptical orbit.When it reaches point A, its engines are turned on and its velocity is suddenly increased. Determine the required increase in velocity so that it enters the second elliptical orbit.
44
p. 165, 13-132The satellite is in an elliptical orbit having an eccentricity of e = 0.5. If its velocity at perigee is vP = 15 Mm/hr , determine its velocity at apogee A and the period of the satellite.
46
47